Trump and UK Prime Minister May talk Iran, Brexit, trade at joint news conference

May is set to step down as her party's leader at the end of this week.

U.K. Prime Minister Theresa May stressed the importance of the U.S.- U.K. "special relationship" in dealing with Iran, but noted "we stand by the nuclear deal" at a joint news conference on Tuesday with President Donald Trump.

"We can also differ sometimes on how to confront the challenges we face," May said as both leaders noted the importance of their alliance as the 75th anniversary of the D-Day invasion approached.

“I’ve always talked openly with you, Donald, when we have taken a different approach and you've done the same with me,” May said to Trump, also highlighting their differences on dealing with climate change on the second day of his visit.

Trump, who withdrew the U.S. from the Iran nuclear deal a year ago, said the two nations must keep Iran from obtaining a nuclear weapon. "I believe that will happen," said Trump.

The president also was optimistic about a trade deal between the two countries in the wake of Brexit, which he said "should happen."

"We're going to have a great and comprehensive trade deal," the president said. Responding to reports that the U.S. wants private American companies to compete with the U.K.'s National Health Service, Trump said, “Look, I think everything with a trade deal is on the table. When you're dealing on trade everything is on the table. So NHS or anything else. A lot more than that. But everything will be on the table, absolutely.”

May quickly jumped in to say the U.K. would negotiate a trade deal only in the nation's best interest.

May pointedly noted that she had declined Trump's advice to sue over Brexit and still had gotten a good result, although her failure to secure a deal with the European Union has led to her being forced out as prime minister. She leaves office on Friday.

"I would have sued and settled," Trump responded. "She's probably a better negotiator than I am," he continued. "I think you deserve a lot of credit," he said, in contrast to his earlier harsh criticism of how May had handled the issue. He called her "tremendous" as both leaders repeatedly complimented each other in a friendly exchange despite their differences.

Delving into U.K. domestic politics when asked about his reported phone call with Boris Johnson, a possible successor to May as prime minister, Trump said, "I know Boris, I've liked him for a long time, I think he'd do a good job. "

Trump said he hadn't seen much in the way of protests, including one featuring a giant "Trump baby" balloon. One was led by Labour Party leader Jeremy Corbyn, whom Trump said he had turned down when he requested a meeting. Trump said he had seen only a "small" protest," calling news reports about large protests"fake news."

He also called London's mayor, Sadiq Khan, with whom he has a running feud, "a negative force, not a positive force" who "hurts the people of this great country."

Later Tuesday, at Winfield House, the U.S. ambassador's residence, Nigel Farage, the leader of the newly formed Brexit Party and one of the chief advocates for Britain leaving the European Union, arrived to meet with Trump.

Before Trump arrived in the U.K., he gave an interview to The Times in which he said Farage should be sent in to negotiate a no deal Brexit.

Farage tweeted later, "Good meeting with President Trump – he really believes in Brexit and is loving his trip to London."

On Trump's threatened tariffs on Mexico, the president pushed back on reports congressional Republicans might try to block them.

"I think, if they do, it's foolish, Trump said. "Mexico should step up and stop this invasion of our country," he added, saying he would carry out his threat to impose the tariffs beginning next Monday, June 10.

Trump also said the U.S. and U.K. would continue to share intelligence despite their differences over how to treat the Chinese tech company Huawei, which the U.S. views as a spying threat.

The two leaders came from a meeting earlier at 10 Downing Street where they continued discussions about a post-Brexit trade deal.

First lady Melania Trump looked on from the first row and other Trump family members, Ivanka Trump, Donald Trump Jr, Eric and Lara Trump and Tiffany Trump watched from the second row.

"I don’t know exactly what your timing is but stick around let’s do this deal," Trump said to the outgoing prime minister -- seemingly in jest -- in remarks before cameras at the beginning of a roundtable with U.S. and U.K. businesses.

The second day of Trump's state visit was being filled with more official business than ceremony, with his meeting with May and business leaders to discuss trade and other matters.

The president’s visit comes at a particularly awkward moment for May, who announced her resignation last Friday and is set to officially step down as the Conservative Party leader on June 7, just a few days after the president’s visit.

The day began with business. President Trump attended a business breakfast meeting at St. James’s Palace co-hosted by May that brought together major U.S. and U.K. business leaders.

The president expressed confidence that the two allies will ultimately reach a "substantial" and "fair" trade deal.

"I think we’ll have a very very substantial trade deal, it will be a very fair deal," Trump said. "This is something my folks want to do, your folks want to do, and we’ll get it done."

In the meeting with U.S. and U.K. businesses, the president praised the strong trade relationship that already exists between the two countries and predicted that there is a "great opportunity to tremendously expand that relationship."

Trump also praised what he said has been an "outstanding" relationship with May, saying he "very much appreciate[s] the relationship we’ve had."

Yet, the persistent uncertainty around Brexit loomed at the breakfast. While the president has expressed his hopes for negotiating a bilateral trade agreement with the U.K., the president’s National Security Adviser John Bolton said the U.S. is waiting for the U.K.’s planned exit from the EU to occur first.

“President Trump remains very eager to cut a bilateral trade deal with an independent Britain. It’s what the people voted for in 2016, and when they get out, whether it’s now, April 12 or later, we’ll be standing right there waiting for them,” Bolton said in a recent interview with Reuters.

In the evening, the president and first lady reciprocated the hospitality of their British hosts with a dinner hosted at Winfield House, located in Regency Park. Prince Charles and Camilla attended the dinner in the queen’s place, according to the palace. The president and first lady were formally dressed, the president in a black tie and first lady in a bright red cape-style gown.

The Trumps greeted Prince Charles and the duchess with a smile and handshakes when their car approached and the two couples posed for photos before the cameras. The president appeared to make a little small talk with the prince as they posed.

The president and first lady entertained guests at separate tables with white roses and candles. The president was seated between the Prince of Wales and Prime Minister May, while the first lady was seated next to the Duchess of Cornwall and Mr. May. Guests including Treasury Secretary Steven Mnuchin and former Prime Minister Tony Blair dined on a menu that included "heritage tomatoes with fresh burrata," "grilled fillet of beef," and summer berries with ice cream.

ABC News' Zoe Magee, Liz Alesse, and Meridith McGraw contributed to this report.