Boris Johnson, a pro-Brexit former foreign secretary and former mayor of London, is the odds-on favorite to replace her — a prospect that clearly delights Mr. Trump, who all but endorsed him last month during his visit to Britain. Mr. Johnson has pledged to pursue a “no deal” exit from the European Union.

“The good news for the wonderful United Kingdom is that they will soon have a new prime minister,” Mr. Trump said. “While I thoroughly enjoyed the magnificent State Visit last month, it was the Queen who I was most impressed with!”

Mr. Trump praised the queen as a “spectacular woman” and claimed that during their meetings, her staff said she had not had so much fun in 25 years. By contrast, he and Mrs. May joked uneasily about their differences over how Britain handled its negotiations with the bloc.

“I seem to remember the president suggested that I sue the European Union,” Mrs. May said. “Which we didn’t do. We went into negotiation, and we came out with a good deal.”

“I would have sued, but that’s O.K.,” Mr. Trump replied. “I would have sued, and settled, maybe.”

In his cables, Mr. Darroch described the White House as a “uniquely dysfunctional environment” and said Mr. Trump was an unpredictable character. “There is no filter,” he wrote. He also said the president faced the prospect of further disclosures that could lead to “disgrace and downfall.”

The cables were obtained by a British tabloid, The Mail on Sunday, which published them over the weekend. The leak prompted theories about whether pro-Brexit forces were less interested in harming Mr. Darroch, whose tenure in Washington is nearing an end, than in torpedoing a likely successor: Mrs. May’s national security adviser, Mark Sedwill. Mr. Sedwill is viewed by some in London as having maneuvered to prevent the government from pursuing a no-deal Brexit.

For all of his private criticism, Mr. Darroch has cultivated close ties with people in Mr. Trump’s orbit. He hosted lavish diplomatic parties at the ambassador’s baronial residence that drew a parade of prominent officials, including Ms. Trump and her husband, Jared Kushner; a former Trump White House chief of staff, John F. Kelly; the former White House chief strategist, Stephen K. Bannon; and Mr. Trump’s senior adviser Kellyanne Conway.