LONDON — Theresa May is heading off the world stage, the curtain on her role as British prime minister about to close. On Tuesday, she had what may be a last hurrah, standing at the twin lecterns with President Trump, who promised that he could deliver a “phenomenal trade deal” with Britain.

But that deal — whatever it is — would be struck with May’s successor. There is no sign that this agonizing, divisive thing called Brexit will be over anytime soon. It could be months, or it could be years. And there can be no new trade deal between the United States and the United Kingdom as long as Britain remains a member of the European bloc.

At their joint news conference, May seemed the bit player. The questions were not really directed at her, because she is soon gone. She steps down as leader of her Conservative Party on Friday and will serve as a kind of caretaker prime minister into the summer, while her many rivals campaign to replace her.

Trump took pains to praise May as “a tremendous professional and a person that loves your country dearly.”

May went on a charm offensive immediately after Trump’s election. She was the first foreign leader to meet him in the Oval Office — a meeting in which she began to push a post-Brexit trade deal. But these are two leaders with very different styles and worldviews, and they never developed a special relationship.

On Tuesday, although much gentler than in the past, Trump reasserted his view that May should have handled Brexit differently.

May noted, “I seem to remember the president suggested that I sued the European Union, which we didn’t do. We went into negotiations, and we came out with a good deal.”

Trump parried: “I would have sued, but that’s okay. I would have sued and settled, maybe, but you never know.”

He then changed course a bit: “She’s probably a better negotiator than I am.”

“Perhaps you won’t be given the credit that you deserve if they do something, but I think you deserve a lot of credit,” he said.

A year ago, on his first official trip to Britain, Trump surprised — actually, shocked — 10 Downing Street by trashing May’s Brexit plan in an interview with the Sun tabloid on the eve of his arrival. “I would have done it much differently,” Trump asserted. “I actually told Theresa May how to do it, but she didn’t listen to me.”

News of that interview last summer broke as May was feting Trump and his wife at Blenheim Palace, birthplace of Winston Churchill. Guests reportedly were staring at their mobile phones and gasping.

Brexit has been May’s undoing. It has consumed her premiership and crushed her. She is being ousted not by the voters or the opposition, but by her own Conservative Party for failing to deliver Britain’s divorce from the E.U. as promised.

Earlier and more vocal support from Trump of a post-Brexit U.S.-U.K. trade deal wouldn’t have saved her. But his criticism that her compromise approach to Brexit would “kill” prospects for a future trade deal didn’t help.

On Tuesday, Trump brushed aside concerns over other looming divisions between the two countries, including over Britain’s possible embrace of Chinese Huawei telecommunications, which the United States has branded a national security risk. “We’re going to have absolutely an agreement on Huawei and everything else,” he said. “We have an incredible intelligence relationship and we will be able to work out any differences.”

The forging of the great 20th-century alliance during and after World War II is a theme of Trump’s Europe trip this week. It is an alliance that many Europeans believe Trump is threatening.

Trump will attend ceremonies in the British port city of Portsmouth on Wednesday to honor Allied forces who sailed from there for the D-Day invasion. On Thursday, he will be in Normandy, France, for the 75th anniversary of that invasion.

He and members of his family were given a tour Tuesday of the underground bunker where Churchill led the country as prime minister during World War II. The scene has been featured in recent films depicting Churchill and is a popular tourist site.

Even with all the spectacle associated with Trump’s state visit, which on Monday included a banquet hosted by the queen at Buckingham Palace and tea with Prince Charles, the British public and press are very much focused on their domestic political mess.

At the news conference with May, Trump was asked about her various potential successors. And he could not resist a little political handicapping. He mentioned knowing two of the contenders — Boris Johnson and Jeremy Hunt — and said he thought both would “do a very good job.”

On June 4, President Trump and Prime Minister Theresa May reinforced their support for Brexit and weighed in on two Conservative Party politicians. (The Washington Post)

British media reported that Johnson turned down a meeting with the president with the excuse that he had a conflicting event. Instead, the two had a “friendly and productive” 20-minute phone call, according to the Press Association.

Trump met in person with other hardcore euroskeptics on Tuesday, including Nigel Farage, whose new Brexit Party crushed all others in Britain’s elections for European Parliament last month. Farage tweeted: “Good meeting with President Trump — he really believes in Brexit and is loving his trip to London.”

Outside these friendly sessions, in the streets of London, tens of thousands protesters gathered to contest Trump and his policies on climate change, migration and other issues.

During June 4 protests against President Trump's visit to London, a 16-foot-tall, animated "robot" of Trump on a toilet was featured in Trafalgar square. (Press Association)

Activists estimated 75,000 people hit the streets Tuesday — fewer than those at the anti-Trump rally in 2018, which organizers said drew more than 100,000.

But the protesters were vocal, their chants — including “Say it loud, say it clear! Donald Trump’s not welcome here!” and “Donald Trump, shame on you!” — ringing in the air as reporters headed to the news conference.

Trump dismissed the demonstrations, which included a talking “Trump Robot” seated on a toilet, saying they were small and insignificant. During the news conference with May, he falsely claimed thousands of people had turned out to cheer him on Monday and Tuesday, and called coverage of the protests against him “fake news.”

“Even coming over today, there were thousands of people cheering, and then I heard that there were protests. I said, ‘Where are the protests? I don’t see any protests.’ I did see a small protest today when I came. Very small.”

Trump’s motorcade passed near the huge orange balloon depicting him as an angry baby that debuted during a less formal working visit last year.

A giant inflatable balloon depicting President Trump as a snarling baby flew outside the British Parliament in London on June 4. (Reuters)

Trump caused a stir by saying all aspects of commerce, including Britain’s National Health Service, should be “on the table” for trade discussions with Washington.

The National Health Service is a postwar creation treasured by Britons of all political stripes, even as they complain about the wait times. One of the most resonant arguments in the Brexit campaign was that by leaving the European Union, Britain could take back control of its money and invest in the health system. After all that, the notion that U.S. companies could come in and bid on the service touched a nerve.

It immediately became a talking point among those vying to succeed May. “The NHS is not for sale to any country and never would be if I was Prime Minister,” former Brexit secretary Dominic Raab tweeted.

Trump may not have realized what he was getting into. British broadcaster Piers Morgan reported that in an interview with the president that airs Wednesday, Trump backtracked, saying, “I don’t see it being on the table. That’s something I would not consider part of trade. That’s not trade.”

Trump also talked up future trade prospects during a Tuesday morning roundtable of American and British business executives.

“I think we will have a very, very substantial trade deal,” he said. And then, leaning toward May, he suggested: “I don’t exactly know what your timing is, but stick around. Let’s do this deal.”

During what was billed as a business roundtable, President Trump asked Prime Minister May to "stick around." May announced her resignation May 24. (The Washington Post)

When asked at the afternoon news conference whether she would take Trump’s advice on that score, May demurred: “Nice try, but no. Look, I’m a woman of my word.”

The scene during President Trump’s visit to Britain

June 5, 2019 | World leaders look on during a commemoration of the 75th anniversary of D-Day in Portsmouth, England. Front row from left: French President Emmanuel Macron, British Prime Minister Theresa May, Prince Charles, Queen Elizabeth II, President Trump, first lady Melania Trump, Greek President Prokopis Pavlopoulos and German Chancellor Angela Merkel. (Chris Jackson/AP)

Karla Adam and Toluse Olorunnipa in London and Marisa Bellack and John Hudson in Washington contributed to this report.