President Trump spent his second day of his visit to the United Kingdom reaffirming the U.S.-U.K. alliance alongside British Prime Minister Theresa May, meeting with Brexit Party leader Nigel Farage, denying the existence of large protests against his visit, and dining with Prince Charles and May.

Mr. Trump and May restated the alliance of their two nations in a news conference Tuesday, even as the president repeated criticisms of some British leaders. Politics at home also came up -- Mr. Trump reiterated his threat to impose tariffs on Mexico and suggested that the GOP will not act to block the tariffs. Doing so, Mr. Trump said, would be "foolish."

The joint news conference in London came just three days before May's resignation is to take effect.The prime minister was forced to announce last month that she would step down after parliament refused three times to back her proposed plan for the U.K.'s withdrawal from the European Union.

On Wednesday, the president will visit Portsmouth with May and the first lady to commemorate D-Day.

Trumps dine with May and Prince Charles

Prime Minister Theresa May, Prince Charles of Wales, and Camilla, the duchess of Cornwall, dined at Winfield House with the president and first lady.

Most of the dinner was closed to the press, but reporters were allowed inside for a brief moment to see Mr. Trump sitting next to Prince Charles and May. Also seated at that table next to Prince Charles was White House press secretary Sarah Sanders.

Here is the menu for the dinner, provided by White House pool reporters:

Nigel Farage spotted leaving residence where Trump is staying

Brexit Party leader Nigel Farage was spotted leaving Winfield House, the residence of the U.S. ambassador where Mr. Trump is staying, around 5:30 p.m. local time, 12:30 p.m. Eastern. The White House has not yet offered further details.

Farage confirmed the two met in a tweet.

Mr. Trump had told reporters when he left Sunday night that he might meet with Farage while in London.

Well I may meet with him, he's been a friend of mine he's been very nice," the president told reporters on the White House South Lawn Sunday night. "I have a very good relationship with him. I have a very good relationship with Nigel Farage, with many people over there and uh, we'll see what happens. But I may meet with him. They want to meet but we'll see what happens."

Trump's adult children on the state visit

Mr. Trump's adult children who are not employed by the administration -- Eric Trump, Donald Trump Jr., and Tiffany Trump -- have been present for much of the state visit and had front seats at the Trump-May press conference as well. The White House has not offered an explanation as to why the president's children who have no role in the administration are on the trip.

Donald Trump Jr. is vocally very supportive of his father, and Eric Trump's wife has worked on the president's reelection campaign. Tiffany Trump is a law school student at Georgetown University. It's also unclear who paid for their travel costs and associated expenses.

Trump says it would be "foolish" for Republicans to block Mexico tariff threat

Mr. Trump insisted Mexico still isn't doing enough to halt illegal immigration, and said the tariffs will be imposed next week.

He also suggested Republicans won't go through with a possible plan to block the tariffs, as reported by the Washington Post. Doing so, Mr. Trump said, would be "foolish."

"I'm a woman of my word," May says on whether she will stick around longer

May, asked if she might stick around longer to work out a trade deal with the U.S., said she will be leaving as planned.

"I'm a woman of my word," she said.

Trump says U.S. and U.K. will "be able to work out any differences"

Mr. Trump, asked if he would consider limiting intelligence sharing with the U.S. over differences on things like the Chinese company Huwaei, insisted that won't be necessary. The U.S. and U.K. "be able to work out any differences," Mr. Trump said.

The president went on to say he has confidence that a trade deal will be reached between the U.S. and U.K.

The president also praised the way May has handled Brexit, despite past criticism.

"Well I don't like to take positions in things that I'm not you know really -- I understand the issue really well I really predicted what was going to happen," Mr. Trump said of the future of Brexit. "Some of you remember that prediction, it was a strong prediction, made at a certain location at a development we were opening the day before it happened. And I thought it was going to happen because of immigration more than anything else but probably it happens for a lot of reasons."

"But I would say I would think that it will happen and it probably should happen. This is a great great country and it wants its own identity. It wants to have its own borders, it wants to run its own affairs. This is a very very special place and I think it deserves a special place. I thought maybe for that reason and for others but for that reason it was going to happen. Yeah I think it will happen."

Trump inaccurately calls protests "fake news"

The president then went on to say many reports of protesters are "fake news," insisting that the only protesters he saw were a very small group.

"There were thousands of people cheering," the president insisted.

"I did not see the protesters until just a little while ago and it was a small group of people put in for political reasons," the president said.

But thousands of protesters did show up on London's streets Tuesday to voice their disapproval of the president's visit and policies.

Trump responds to criticism from British leaders

In the first question, the president was asked to respond to criticisms from top leaders in Britain, including Jeremy Corbyn and London Mayor Sadiq Khan.

"Well I think he's been a not very good mayor from what I understand," the president said of Khan, adding he doesn't think Khan should be criticizing a representative of the United States.

"I think he should actually focus on his job," the president said.

On Corbyn, Mr. Trump said he denied a meeting with the leader of the Labour Party, calling him a "negative force."

Trump thanks U.K. for longstanding relationship

Mr. Trump then began his remarks, describing the queen as a "fantastic" person and thanking her for her hospitality.

The U.S. and U.K., the president said, have a special relationship, from D-Day to today -- "a liberation like few people have seen before" -- to Britain's aid in attempting to defeat ISIS.

Mr. Trump said the U.S. is "committed to a phenomenal trade deal" with the U.K., as the U.K. looks to implement Brexit.

The president also praised May for her love of her country and partnership.

Trump and May take the stage

Mr. Trump and May took the stage at 9:07 a.m. Eastern time, with May beginning remarks first.

"This week we commemorate the extraordinary courage and sacrifice of those who gave their lives for our liberty on D-Day, 75 years ago," the prime minister said, noting the "special" relationship between the U.S. and U.K.

May went on to describe specific key moments in the U.S.-U.K. alliance. She also noted her appreciation for Mr. Trump's push to have NATO members pay more towards their mutual security.

The outgoing prime minister said the two discussed the importance of pressuring Iran, "although we differ on the means of achieving that," she noted.

The Trump administration has acted largely alone in intensifying pressure on Iran and backing out of the Iran nuclear deal last year.

Joint press conference beginning shortly

Both British and American flags adorn the stately room.

While reporters are sure to ask questions about the U.K.-U.S. relationship, anything is possible in these news conference settings with a president who likes to speak his mind.

First Lady hosts garden party at 10 Downing​

First Lady Melania Trump hosted a reception with the prime minister's husband, Philip, for 10 Downing and British embassy families during the president's day of meetings at the official residence. There was a mix of British and American lawn games and food at the lighthearted event, breaking up a day of political talks between the two nations.

The first lady met with young children waving British and American flags and signed a banner commemorating the state visit.

Sadiq Khan calls Trump a "poster boy" for far right

"I'm a bit surprised that the President of the USA would, frankly speaking, behave like an 11-year-old and resort to name calling," London Mayor Sadiq Khan told CBS News partner network BBC News when reacting to Mr. Trump's tweets about him. Khan said despite their war of words, he'd welcome a "discussion about some of the issues we disagree about."

But Kahn said he was concerned that the leader of the U.S. "seems a poster boy for far right movements across the globe" suggesting his rhetoric and past ban on travel from Muslim-majority countries "played into the hands of extremists."

Khan urged the prime minister to express to Mr. Trump during their day of meetings "her concerns in relation to some of the things he's said and done."

Trumps, Mays view copy of Declaration of Independence

During a tour of 10 Downing, Prime Minister Theresa May, Philip May, President Trump and the first lady viewed a copy of the Declaration of Independence.

Thousands protest as Trump arrives for May meeting

As the Trumps arrived, an audible hum of chants rang out in the distance as thousands gathered in London's Trafalgar Square, just a 3 minute drive down the street from 10 Downing.

Crowds of protesters were seen with signs demonstrating against the Trump administration while a giant balloon of the president wearing diapers dubbed "Trump Baby" soared overhead. The anti-Trump protests are expected to continue throughout much of the day as the president concludes his State visit.

Trump arrives at 10 Downing

The President and first lady arrived at 10 Downing Street and were greeted by Prime Minister May and her husband Philip May. The group exchanged pleasantries and posed for photos before entering the building for private meetings.

Staff at the residence rolled out the red carpet for the president as Larry the cat, a common presence outside 10 Downing Street, was seen resting in a window sill during the welcome ceremony.

White House adviser Jared Kushner was seen arriving several moments before the president entered the building. White House officials including the president's daughter Ivanka Trump, Press Secretary Sarah Sanders, Acting Chief of Staff Mick Mulvaney, Treasury Secretary Steven Mnuchin and National Security Adviser John Bolton arrived for the day's events shortly after the president entered. Secretary of State Mike Pompeo is also slated to attend.

The president is now expected to participate in an expanded bilateral meeting with the Prime Minister.

Business round table

Mr. Trump sat opposite Prime Minister Theresa May at one of the royal palaces in London on Tuesday, each leader flanked by economic advisers and company bosses from some of the biggest businesses on both sides of the Atlantic.

According to the AFP news agency, bosses from corporate giants BAE Systems, GlaxoSmithKline, Barclays, Reckitt Benckiser, JP Morgan, Lockheed Martin and Goldman Sachs International were among those expected to attend the meeting.

Speaking before his departure from Washington, Mr. Trump said Britain's leaders "want to do trade with the United States and I think there's an opportunity for a very big trade deal at some point in the near future."

The U.K.'s scheduled departure from the EU may leave it free to strike its own unilateral trade deals around the world, but it will also leave it without the huge collective bargaining power of membership in the 28-nation trade bloc. In an interview with British newspaper The Sunday Times, Mr. Trump said the U.K. should "walk away from Brexit talks" if the European Union doesn't give it what it wants.

"We are your largest partner. You're our largest partner. A lot of people don't know that," Mr. Trump said to May as the meeting began, noting that he believed there was a "great opportunity to enlarge that, in light of what's happening."

Mr. Trump's declaration got the facts ironically wrong, however. The U.S.'s biggest trading partner is the European Union collectively, not Great Britain on its own, which is only the second largest economy in the trading bloc behind Germany.

But what Mr. Trump's administration might demand from Britain in exchange for a trade deal has already proven controversial in the U.K.

To strike a trade deal with the much larger U.S., the Trump administration has indicated that Britain may be asked to open its cherished national health care system, the NHS, up to American investors. Opposition Labour Party lawmakers were quick to point to Ambassador Woody Johnson's remarks as evidence that the Conservative-led government was willing to "sell" Britain's social healthcare system.

Johnson told the BBC's Andrew Marr Show that, "in a trade deal, all things that are traded would be on the table." Asked by the host if that specifically meant healthcare, Johnson replied: "I would think so."

Johnson also appeared to confirm fears among some Britons that in order to strike a deal with the U.S., post-Brexit Britain might need to lower some of the food safety standards in place under EU law, which ban genetically modified organisms and some pesticides and practices used widely in U.S. agriculture and food production.

"There will have to be some deal where you give the British people a choice," he told Marr's audience. "American products can come over and be allowed to come over. Agriculture is extremely important to the president and to any American president ... but if the British people like it, they can buy it, if they don't like it, they don't have to buy it."

Protests planned for Trump's visit

The president's trip to the U.K. has been met with protests, and the president has delivered some fiery words of his own. Anti-Trump protests were planned Tuesday in London, including a robotic "Trump" sat upon a toilet that will be paraded around the city -- along with the infamous "Trump baby" balloon.

Mr. Trump's spat with London Mayor Sadiq Khan has played out on Twitter. On Monday, the president compared him to one of his arch rivals, Bill de Blasio.

"I don't think much of him," the president said of Khan Sunday night. "I think that he's the twin of de Blasio, except shorter." Mr. Trump called Khan a "stone cold loser."

Khan criticized Mr. Trump in 2016 for his views on Islam, calling Mr. Trump "ignorant." At the time, Mr. Trump responded by questioning Khan's IQ.

Despite low approval ratings in the U.K., the president tweeted that he thought the media would have to work "hard" to find people demonstrating against his presence in the British capital. Over the past couple days, a handful of demonstrators have popped up in London wearing Mr. Trump's trademark "MAGA" hats, supporting his visit.

One British man, wearing the hat near the protesters as they inflated "Trump baby" on Tuesday, told CBS News' Haley Ott that he hoped to encourage people to have, "a bit of civil discourse, find some common ground."

Trump's visit so far has been ceremonial

The first day of Mr. Trump's visit to the U.K. was largely ceremonial, and the White House has laid out few objectives for his first official state visit to Britain.

Mr. Trump received a Winston Churchill book from Queen Elizabeth II, with whom he also had lunch on Monday. He then had tea with Prince Charles and his wife, Camilla. On Monday night he took part in a state banquet at Buckingham Palace, during which he and the queen delivered speeches.

"Mr. President, as we look to the future, I am confident that our common values and shared interests will continue to unite us," the queen said. "Tonight we celebrate an alliance that has helped to ensure the safety and prosperity of both our peoples for decades, and which I believe will endure for many years to come."

For his part, the president expressed gratitude for the queen's hospitality and remembered those who lost their lives on D-Day, the pivotal Allied invasion of northern France during World War II, the 75th anniversary of which is on Thursday.

"The bond between our nations was forever sealed in that great crusade," the president said in his speech. "As we honor our shared victory and heritage, we affirm the common values that will unite us long into future."