Jeremy Hunt, the foreign secretary, challenged Boris Johnson, his predecessor, to say he would resign as prime minister if he did not extract Britain from the E.U. on Oct. 31, as he has promised. Mr. Johnson dodged, but he also went on the attack, saying he admired Mr. Hunt’s ability “to change his mind” — referring to his former opposition to Britain’s withdrawal from the bloc.

Trump factor: Mr. Trump has laid into the British ambassador, Kim Darroch, calling him “wacky” and “a very stupid guy,” over leaked internal documents in which Mr. Darroch derided the White House.

Mr. Hunt described Mr. Trump’s comments as “unacceptable,” while Mr. Johnson offered only mild criticism and said he had good relations with the White House.

Markets: The pound sterling neared a two-year low on Tuesday as the diplomatic spat raised questions about what kind of support a post-Brexit Britain can expect from the Trump administration.