Mr. Carney, who was due to leave the job in 2018, agreed twice to extend his term as the Brexit process ground on, working toward monetary and financial stability as the rupture with Europe threatened the very economy he was aiming to keep steady.

But now, as Mr. Carney’s Jan. 31 departure approaches, the appointment of the bank’s next governor is entangled in the most political of issues: Who will be running the British government in a few months and managing Brexit?

Under former Prime Minister Theresa May, Britain’s government started the search for a successor earlier this year. But there was no decision before Mrs. May lost her way in the Brexit battle and was succeeded by the current prime minister, Boris Johnson.

The Treasury put a call out for applicants in April and maintains that it expects to make an appointment in the fall. That choice would require the prime minister’s blessing and then a review by a parliamentary committee. But Parliament continues to be torn by Brexit, leaving little opportunity for any significant committee work before the departure date of Oct. 31.

[A look at how Brexit could break Britain’s food chain.]

Mr. Johnson’s tactics to push on with Brexit — even expelling members of his own party who disagreed in the last few weeks — has roiled all the political parties, with some members of Parliament calling for a new election. This has added more uncertainty to the selection of the bank chief and raised the question of whether the current government will even be in charge.