Theresa May’s Brexit deal could lead to electoral armageddon for the Conservative party, three leading Tory rebel MPs and former ministers have warned as they threw their weight behind a “people’s vote”.

In his first major speech since resigning as transport minister earlier this month, Jo Johnson joined David Willetts and Justine Greening to sketch out a future in which their party faced an existential crisis and would have its brand thrashed by the economic fallout from Brexit.

The package their party leader had agreed with the EU was described by Johnson as a “botched deal” that would put British firms at a competitive disadvantage and fail the services sector, which he said had been “scandalously” neglected during negotiations on Brexit.

He added: “Brexit is seen as a project driven by the Conservative party and this half-baked, worst of all worlds Brexit could trigger an electoral defeat on the scale of 1997, or worse, with this ‘Tory Brexit’ label an albatross around our necks for years to come.”

Such an outcome would “roll out the red carpet for Jeremy Corbyn” and lead to what he described as “communist ideologues” coming to power in Britain.

Johnson also sounded a warning about the potential breakup of the UK by expressing concern that the current deal treated Northern Ireland in a different way from Britain, heralding a huge extension of regulatory checks in the Irish Sea that could shift trade patterns and block British goods out of Northern Ireland.

“Unionists across our four nations are right to be worried,” he said. He added that Scottish nationalists would use the different treatment of Northern Ireland as a grievance and push for independence.

Name-checking his brother Boris and other senior Brexiters such as Dominic Raab, who he said were warning that May’s deal was worse for the UK than actually staying in the EU, Johnson said that he could understand their anger.

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To laughter, he added: “When we were told that Brexit would mean taking back control, none of us in our wildest imaginations thought it would mean powers being taken [from Westminster] to other European parliaments and the European parliament.”

Johnson was introduced by the veteran former minister David Willetts, who said the trio were seriously concerned about the threat Brexit posed to their party and, in particular, its reputation for economic competence and as a guardian of living standards.

The three appeared at an event at the Southbank Centre in London organised by the People’s Vote campaign – which brings together cross-party remain-supporting MPs – where Willetts said they were presenting the “Conservative case” for such a poll.

“When we came into government in 1979 and again in 2010 we came in and we sorted out a mess,” said Willetts. “Now we are in danger of imposing declining living standards on the British people, and the Conservative party will have to take responsibility for that.”

Addressing an audience of journalists and young activists associated with the campaign for a second vote on Brexit, he added: “The future of the party depends on the votes of younger people. Brexit jeopardises that.”

Sketching out a plan for another referendum that would take place by the end of 30 May 2019, Greening said article 50 could be extended by another four months.

The former education secretary said she was backing a people’s vote because the country was tired of what she described as “backroom deals” and said it was now incredible to see May “touring the country and talking to people who she did not want to give a vote to”.