Theresa May claims to have secured significant changes to her Brexit deal in a last-minute dash to Europe just hours before she must put her plan to a critical vote in parliament.

In a late night statement on Monday in Strasbourg she argued the new-look deal meant Britain could not be trapped in the “Irish backstop” so hated by Eurosceptic Tories and her DUP allies, but major doubts remain over whether it is enough to win their backing on Tuesday.

The prime minister’s deputy David Lidington warned that if her deal is rejected for a second time by MPs it will “plunge the country into a political crisis”.

European leaders warned there would be no “third chance”, but Conservative Brexiteers insisted there are still “very worrying features” to the agreement, while Labour leader Jeremy Corbyn said “MPs must reject this deal tomorrow”.

The announcement came after another dramatic day in Westminster on Monday, which began with talk of Ms May potentially delaying Tuesday’s vote on her deal after a seemingly fruitless weekend of talks.

But speaking an hour before midnight, she said: “MPs were clear that legal changes were needed to the backstop. Today we have secured legal changes.

“Now is the time to come together, to back this improved Brexit deal, and to deliver on the instruction of the British people.”

The backstop is an arrangement in the existing withdrawal agreement that comes into play if the EU and UK fail to agree future trading arrangements by the end of 2020, thus keeping the Irish border open, but also locking the UK into a customs union with the EU on a potentially indefinite basis.

Europe Commission president Jean-Claude Juncker said the new additions to the agreement provided “meaningful clarifications and legal guarantees” on the backstop’s temporary nature.

He went on: “In politics sometimes you get a second chance. It is what you do with this second chance that counts – because there will be no third chance.

In politics sometimes you get a second chance. It is what you do with this second chance that counts – because there will be no third chance European Commission President Jean-Claude Juncker

“There will be no further interpretation of the interpretations, and no further reassurances on the reassurances if the meaningful vote fails tomorrow.”

In a commons statement Mr Lidington revealed that the UK had secured two new documents, a “joint legally binding instrument on the withdrawal agreement” and a “joint statement to supplement the political declaration” on future relations.

There is also a third element – a unilateral declaration from the UK setting out what actions it would take if it felt the backstop is being abused by the EU.

Mr Lidington said the new legal “instrument” confirmed that the EU could not try to trap the UK in the backstop indefinitely, because commitments they had made to not do so were now legally binding.

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The cabinet minister argued that this commitment from the EU could also now be used as the basis of a formal dispute through an independent arbitration, which could lead to the suspension of the backstop.

Furthermore, he said the UK had won a commitment to find an alternative arrangement to the backstop by 2020, something set out in the “joint statement”.

Critically the legal instrument looks like it will be enough to allow attorney general Geoffrey Cox to change his legal advice to parliament, which previously said the backstop could be indefinite.

Mr Lidington said: “Tomorrow will be a fundamental choice. To vote for the improved deal or to plunge the country into a political crisis. If we vote for this improved deal, we will both end the current uncertainty and have delivered Brexit.

“The house was clear on the need for legally binding changes to the backstop. Today we have secured those changes.”

To win the vote on Tuesday, Ms May must convince not only backbenchers in the European Research Group, led by Jacob Rees-Mogg, but also Northern Irish MPs in the DUP who prop up her Commons majority.

Irish deputy prime minister Simon Coveney: Theresa May going to Strasbourg for further Brexit negotiations

Both groups have previously called for the backstop to be removed or replaced altogether, saying anything less would be unacceptable, though this position has been softened in some DUP statements and in comments from individuals in the ERG.

But to underline the challenge Ms May faces, the ERG’s deputy-chair Steve Baker said he did not believe Ms May’s changes would go as far as he and many other Brexiteer colleagues had demanded.

He said: “It’s not for the first time that David [Lidington] has had to put a good gloss on something that falls short of what was expected.”

It had looked on Monday morning as if Ms May would enrage all sides of the Brexit debate by trying to either delay or downgrade the vote on Tuesday after failing to win concessions at the weekend.

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Downing Street was forced to deny claims that the EU had made an offer of sorts on Sunday only for it to be turned down in London, but they did reveal Ms May had called a string of EU leaders over the weekend including German chancellor Angela Merkel and French president Emmanuel Macron.

There was also a further call with Mr Juncker, following one that had taken place over the weekend, a further sign of movement.

Then shortly after lunch Irish foreign minister Simon Coveney said he believed Ms May would travel across the channel for discussions, though Downing Street still refused to confirm it.

Meanwhile, Ms May attended a ceremony to commemorate Commonwealth Day at Westminster Abbey in which she read out a bible passage from Corinthians which began “the body is not made up of one part but of many”, which some saw as a message to Tory rebels to get behind her.

It was at about 5pm, as she headed to the airport, that Number 10 let it be known she was finally going to Strasbourg though they continued to play down suggestions that a deal had been struck.

Things suddenly began to move more quickly, with the DUP’s leader Arlene Foster travelling to London for a meeting with her party’s MPs, while the Irish cabinet were also called in for a meeting, as a spokesman confirmed the Taoiseach had also spoken to Mr Juncker.

Guy Verhofstadt, the European Parliament’s Brexit coordinator, confirmed he and the parliament’s president Antonio Tajani also met Ms May on Monday night.

The face-to-face meetings took place in the Strasbourg parliament’s Winston Churchill building, while in London Ms May’s senior ministers were called into the cabinet office to be briefed by Mr Liddington ahead of his statement.

Mr Corbyn said: “The prime minister’s negotiations have failed. This evening’s agreement with the European Commission does not contain anything approaching the changes Theresa May promised parliament, and whipped her MPs to vote for.