May’s top lieutenants acknowledged publicly that Bercow’s unexpected announcement means she will probably go mostly empty-handed to a crucial E.U. summit, starting Thursday, where the remaining 27 E.U. leaders will mull how long to let Britain stay in the club beyond this month. May might propose a time frame in advance: One plan floated in the British media was a long delay of at least nine months, unless she can pass the current divorce deal before July, when a new European Parliament is scheduled to sit.

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“For my Brexit colleagues, I think they can see that there is a growing risk of no Brexit,” Brexit Secretary Stephen Barclay told the BBC. He said May might still try to hold another vote before the Brexit deadline but noted that either way, she will have to seek an extension from fellow E.U. leaders.

On Monday, Bercow invoked a parliamentary convention dating to 1604 that he said banned putting a measure to a vote in Parliament if it has already been rejected in the same session of the legislature. May had hoped Brexiteer lawmakers would be sufficiently spooked by the prospect of getting stuck in the E.U. to pass her plan on the third try.

One way to short-circuit Bercow’s injunction would be to flicker the light switches in Parliament, declaring an end to the current session and starting a new one, legal experts said. But that would require summoning Queen Elizabeth II for some ceremonies, a step Barclay said the leaders would not force her to take. (She turns 93 next month, after all.)

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The spinning of wheels in London left European leaders in disbelief at a process many of them already felt had hit rock bottom. Some reiterated their need for a concrete plan from May about what she would do with the extra time. To grant a long extension, many E.U. policymakers said, they would want assurances that Britain will change its bargaining red lines or hold another referendum or election to sort itself out.

“I will fight until the last hour of March 29 so that we still come to an orderly exit,” German Chancellor Angela Merkel told reporters in Berlin. “I’ll admit that I wasn’t actively familiar with the rules of the British Parliament from the 17th century.”

French leaders appeared to be in a tougher mood, embracing a good-cop, bad-cop routine that has developed in concert with Germany on Brexit issues in recent months.

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The British “have said ‘no’ to a no-deal, and they have said ‘no’ to a realistic deal. Now they have to change their mind on one or the other,” France’s Europe minister, Nathalie Loiseau, told reporters in Brussels.

Loiseau recently wrote on her private Facebook page that she had named her cat Brexit because it could not decide whether it wanted to be inside or outside, according to France’s weekly Journal du Dimanche.

Asked about her cat on Tuesday, she said: “I think I need to have a certain sense of humor to deal with Brexit.”

European leaders were not ruling out the possibility of a chaotic, safety-net-free departure for Britain in 10 days — a step that could snarl trade between Britain and the rest of the world and trigger an economic earthquake.

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“Voting against ‘no deal’ does not prevent it from happening,” the chief E.U. Brexit negotiator, Michel Barnier, told reporters in Brussels. “Everyone should now finalize all preparations for a ‘no deal’ scenario.”

But even assuming they grant an extension, which most European policymakers appear to favor, they will confront a thicket of practical problems about how much time to give. The biggest challenge concerns European Parliament elections that are scheduled for the end of May. If Britain decides not to participate, the longest that most European leaders are willing to allow for an extension is until the end of June.

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Longer than that, and the new parliament will be in session. If Britain is still an E.U. member without any lawmakers in its parliament, that violate E.U. treaties, and the parliament’s actions could be subject to legal challenge.

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Britain’s European Parliament members are already packing up to depart. But even if Britain does win an extension and organizes its elections for the Brussels body in May, E.U. policymakers fear it could play a spoiler role in the bloc’s discussions. So long as Britain remains in the club, it has a voice in its decision-making — an oddity that means May will also sign off on a raft of non-Brexit E.U. policies at the summit this week.