LONDON — It began with smiles and holding hands in the White House. Then she told him he was wrong to send a bunch of tweets the way he did. So he told her to mind her own business.

That, roughly, has been the trajectory of the ties — which some Britons call a special relationship — between President Trump and Theresa May, the British prime minister who hurried to Washington to be the first foreign leader to pay an official visit to the new American leader after his inauguration.

And now the relationship may be put to the test once more. The United States ambassador to Britain, Woody Johnson, said on Tuesday that he believed that Mr. Trump would come to London to dedicate the new United States Embassy, which is scheduled to open in January, reviving a debate among Britons about what kind of welcome Mr. Trump might expect.

The presidential trip has been an on-off kind of affair — more off than on, it has sometimes seemed — ever since Mrs. May invited Mr. Trump on behalf of Queen Elizabeth II to pay an official visit.