Prime Minister Theresa May’s prospects of winning a crucial vote on Britain’s withdrawal from the European Union were dealt a severe blow on Tuesday, when the attorney general, Geoffrey Cox, said that last-minute changes in her plan did not fundamentally alter the legal status of the agreement.Mr. Cox’s opinion is likely to be crucial in swaying the votes of pro-Brexit Conservative lawmakers, so his views leave Mrs. May’s chances of winning the vote in Parliament scheduled for Tuesday evening hanging by a thread.The defeat of Mrs. May’s plan could potentially throw Britain into a new period of political uncertainty, one that could lead to Parliament trying to take control of the process from the government in a further diminishing of Mrs. May’s shaky authority.Mr. Cox said the concessions from the bloc did “reduce the risk” of Britain being trapped indefinitely in the so-called Irish backstop — an insurance policy against a hard border in Ireland, and a main issue for opponents of Mrs. May’s deal.But Mr. Cox said that the understanding did not alter the two parties’ rights and obligations. Even under the new terms, Britain would have “no internationally lawful means of exiting the protocol’s arrangements, save by agreement,” he said.Mrs. May had flown to Strasbourg, France, on Monday to negotiate the changes to the withdrawal agreement, hoping to persuade British lawmakers to reverse the overwhelming rejection of the deal they delivered in January.The changes were intended to help bridge the gap on the confusing and thus far intractable issue of the border between Ireland, which is a member of the European Union, and Northern Ireland, which is part of the United Kingdom, but it was unclear whether the alterations would be sufficient.Speaking late Monday in Strasbourg, Jean-Claude Juncker, the president of the European Commission, the bloc’s executive arm, warned that the British could not expect any further concessions. “There will be no further interpretation of the interpretation,” he said. “No further assurances on the reassurances if the meaningful vote fails tomorrow.”The 650-seat British Parliament, which rejected Mrs. May’s previous deal by 230 votes, is scheduled to vote on the revised agreement around 7 p.m. in London. At stake is the fate of the withdrawal agreement and a political declaration that would allow Britain to leave the bloc on March 29 with a standstill transition period, during which very little would change until the end of 2020.Late on Monday, Mrs. May’s de facto deputy, David Lidington, appealed to Parliament to endorse the proposals, arguing that the alternative would be to “plunge this country into a political crisis.”ImagePrime Minister Theresa May of Britain and the president of the European Commission, Jean-Claude Juncker, in Strasbourg, France, on Monday.CreditVincent Kessler/ReutersIf Britain leaves the European Union without an agreement — the so-called no-deal Brexit — the consequences could be dire, with fears of clogged ports and shortages of some food and medicine.British lawmakers have been promised that, if they reject Mrs. May’s plans on Tuesday, they will be offered two more votes later this week. The first vote would allow lawmakers to formally state that they oppose a no-deal Brexit.A majority of lawmakers do not want Britain to leave the European Union without an agreement, and if they reject the idea of departing under a no-deal scenario, they would then have the opportunity to instruct Mrs. May to seek a postponement of the March 29 exit date.The changes offered by Mr. Juncker were designed to reassure pro-Brexit lawmakers who fear that Britain could be trapped indefinitely inside parts of the European Union’s economic rule book and prevented from striking independent trade deals.The new pledges in a “joint instrument” clarify the temporary nature of the Irish backstop provision, which is part of the withdrawal agreement. The European Union has demanded the backstop as an insurance policy to prevent the creation of a hard border between Ireland and Northern Ireland.Mr. Juncker said that the latest legal tweak “complements the withdrawal agreement without reopening it.” That might not be enough for some pro-Brexit lawmakers, who pushed for the backstop to be eliminated completely or for Britain to be given a time limit or the right to abandon it unilaterally.Instead, the changes give a clearer picture of how Britain might be able to exit the backstop if there were an impasse over the matter.One central question on Tuesday was whether the new provisions would win over the 10 lawmakers from Northern Ireland’s Democratic Unionist Party, whose support allowed Mrs. May to form a government but who have opposed her Brexit plan.If they can be persuaded to reverse course, that could in turn help Mrs. May pick up support from Conservative lawmakers who rejected her deal the first time and are looking for political cover before supporting the prime minister on Tuesday.Iain Duncan Smith, a pro-Brexit lawmaker and former Conservative Party leader, told the BBC that a panel of euroskeptic legal experts would consider the plans and determine whether they matched Mrs. May’s claims.But he also appealed for more time to consider the legal niceties. “I certainly think an extra day would help enormously,” he said.The leader of the Democratic Unionist Party, Arlene Foster, said that she was also sympathetic to the idea of a daylong postponement, the BBC reported.Follow Stephen Castle on Twitter: @_StephenCastle.