Nonetheless the collapse is yet another setback for a prime minister who has already suffered a remarkable string of defeats and been hounded by members of her own party to leave. Parliament has voted three times to reject her Brexit plan, which would keep Britain closely tied to the European Union at least until the end of 2020 but then extract it from the bloc’s main economic structures. She said this week that she would attempt a fourth vote on her plan early next month.

Mrs. May had hoped to lure Labour with the prospect that Britain could stay — at least temporarily — in a type of customs union with the bloc, thereby eliminating the need for tariffs and many border checks on goods flowing between Britain and continental Europe.

But that did not prove enough to tempt a Labour Party that favors retaining closer ties to the European Union to protect the economy. It had argued that Mrs. May had not offered enough concessions or effective safeguards to ensure that her successor could not tear up any agreement.

Britons will vote in elections to the European Parliament next Thursday, almost three years after they opted in a referendum to quit the bloc. Mrs. May now has to gird herself for results that could prove disastrous. One recent poll has her Conservatives trailing both the Brexit Party, led by the hard-line Brexiteer Nigel Farage, and the anti-Brexit Liberal Democrats.

But voters are expected to punish the Labour Party too, and Mrs. May’s last, faint hope of pushing her Brexit plan through Parliament is that a shock to the two main parties might galvanize lawmakers into supporting her deal.

Pressure on Mrs. May and Mr. Corbyn to reach a deal increased this month after local elections that were bad for both parties. While the Conservatives lost 1,300 seats in local municipalities, Labour failed to take advantage, shedding around 80 itself.

Though Mr. Corbyn, a lifelong Euroskeptic, argued that a deal needed to be reached, his party’s divisions have made that outcome difficult. A very large faction in the Labour Party opposes Brexit and wants a second referendum on any deal, so any agreement with the Conservatives would have risked straining loyalty to a breaking point.