The most recent indication of Brexit madness surrounded the sudden resignation of Ivan Rogers, Britain’s ambassador to the European Union, who charged in an email to his staff that the government had “muddled thinking” about Brexit and had not understood what the 27 other nations in the bloc would accept in an exit negotiation.

Mr. Rogers, who had been intimately involved in former Prime Minister David Cameron’s vain effort to secure a winning new arrangement for Britain from Brussels, was being challenged or ignored by those in the government who, unlike Mrs. May herself, favored Brexit from the start.

Nor did he get on with her two closest political advisers, Fiona Hill and Nick Timothy. They reportedly resented Mr. Rogers’s views, which somehow made it into the press, that the government was unprepared for the negotiations, overly optimistic about getting a special deal and utopian about how long it would take to negotiate a separate trade agreement with the European Union.

The press wrote happily about a revolt of the “mandarins” — the permanent civil service — over Mrs. May’s leadership, with those opposed to Brexit judging the loss of Mr. Rogers and his expertise a body blow to the country’s future. Brexiters largely considered his decision to quit both welcome and inevitable.

Brexit advocates, like Iain Duncan Smith and John Redwood, dismissed Mr. Rogers’s email as pompous and whiny and pressed for a known Brexiter to replace him in Brussels. David Davis, the Brexiter who heads the new cabinet office called the Department for Exiting the E.U. (Dexeu), allowed his top civil servant, Olly Robbins, to try to downgrade the ambassador’s job so the entire Brussels Embassy would report to him, and not to the Foreign Office.

The Foreign Office successfully fended off Mr. Davis and demanded to keep the Brussels job. That went to Tim Barrow, a former ambassador to Moscow and the current political director of the Foreign Office. Mr. Barrow, a security expert, is considered a sensible, intelligent and safe pair of hands.

Herself a quiet opponent of Brexit, Mrs. May promised to carry out the wishes of the British people as expressed in the June referendum that caused Mr. Cameron to resign only a year after he won a surprising majority for the Conservatives.

But it was the longstanding anti-European minority in the party, enhanced by defections to the pro-Brexit U.K. Independence Party, that forced the referendum on a country with other priorities, like jobs, crime, security and the failings of the National Health Service.

While for most Brexit Conservatives, the main issue was British sovereignty and freedom from European Union rules, regulations, court decisions and fees, the main concern of the 52 percent of Britons who favored Brexit was immigration, Arron Banks, the prime funder of UKIP and one of the Brexit campaigns, said in an interview. He paid for a private poll of 50,000 voters himself.

Behind concerns about security, jobs and the health service, Mr. Banks said, was a widely felt judgment that immigration numbers were too large and out of control, and that such control could only be restored by leaving the European Union.

Therein lies Mrs. May’s dilemma. Diplomats and officials who have had some discussions with her advisers and would not be named because of the confidential nature of those talks say she has two priorities that will limit her negotiating options.

First, vital to Conservatives who consider sovereignty most important is getting Britain out from under the jurisdiction of the European Court of Justice. Second, control over immigration is important to most Brexit voters.

The logic of these two priorities would mean that Britain could no longer be a part of the European Union’s single market for trade in goods and services and freedom of travel and labor. Nor could it be a part of the customs union for goods alone, because that also would mean both paying Brussels and having no ability to strike separate trade deals with China, say, or Washington.

So the only logical future relationship would seem to be a new trade deal in goods and services that Britain would have to negotiate with the rest of the European Union — a negotiation that Mr. Rogers suggested, to Mrs. May’s unhappiness, could take a decade.

In the meantime, he had suggested, Britain should negotiate a transitional agreement that would preserve free trade and would probably look a lot like Britain’s current membership — with the obligations, but without the right to participate in decision-making.

Without such a transition, Britain risks a “hard Brexit,” with considerable damage to its trade and especially to its dominant financial services sector.

But those conclusions are politically unpopular as well, especially with Brexiters like Mr. Davis who believe that the European Union needs Britain more than it needs the European Union, and like Foreign Secretary Boris Johnson, who likes to say that Britain can have its cake and eat it, too, and that Britain can both control immigration and still remain in the single market.

The loss of access to the single market could also have serious implications for Britain’s integrity, since Scotland’s nationalist leaders threaten a new independence referendum if it can no longer trade freely with the European Union.

But logic and politics do not always fit together nicely. So Mrs. May holds her fire, remains silent about her priorities and their costs and simply promises that she will somehow produce a British exit that satisfies everyone.

In the meantime, however, having chided United States Secretary of State John Kerry for his speech criticizing the Israeli government and settlement expansion in what appeared to be an effort to align Britain with President-elect Donald J. Trump, Downing Street is suggesting that it has finally secured a White House meeting for Mrs. May with Mr. Trump sometime in February.

Mr. Trump confirmed the meeting in a Twitter post on Saturday, saying: “I look very much forward to meeting Prime Minister Theresa May in Washington in the Spring. Britain, a longtime U.S. ally, is very special!” Indeed, a Britain out of the European Union is going to need the so-called special relationship with the United States more than ever.