Boris Johnson does not have enough supporters to replace the prime minister, believes Downing Street

Even before Theresa May danced her self-deprecating way on to the stage in Birmingham this week, most of her party were resigned to the fact that however little they may love her, there is little to be gained from chucking her aside before Brexit next March.

One leave-backing cabinet minister told the Guardian that May was a “vehicle for Brexit”, and she was likely to step down shortly afterwards; another suggested the optimal time to replace her would be in 2021, a year before the next general election is due.

And while Boris Johnson drew a huge crowd for his “chuck Chequers” speech, he didn’t go as far as to call on his supporters to depose the prime minister. Downing Street remains convinced that he doesn’t have the numbers, at least for now.

Q&A Boris Johnson timeline Show Hide February 2016

Johnson says in a long-awaited statement that 'after a huge amount of heartache' he will​ back the leave campaign. In January, David Cameron had announced ministers could campaign on either side, and despite being a backbencher, Johnson's view was seen as influential. May 2016 Johnson campaigns around the UK, posing with the now infamous bus saying leaving the EU would provide the NHS with an extra £350m a week – a claim later condemned as a misuse of statistics. 23 June 2016 UK voters take part in the referendum. In the early hours of the following day the results show 52% voted to leave the EU. At a victory press conference hours later Johnson looks shaken. 30 June 2016 After Cameron quits, Johnson announces he will contest the Conservative leadership, pitching himself as a unifier. In a surprise twist, Michael Gove says he will also stand, warning that Johnson cannot provide the leadership needed. This prompts Johnson to use the launch of his leadership bid to say he will no longer stand. 13 July 2016 Theresa May, who took over the leadership unopposed, appoints Johnson as foreign secretary. November 2017 Labour MPs call for Johnson to step down after he wrongly states that Nazanin Zaghari-Ratcliffe, who was jailed in Iran, was 'teaching people journalism' – something both her family and her employer said was untrue, and was used by Iranian officials to justify her detention. 6 July 2018 Johnson is among 29 ministers who discuss May’s plan for Brexit at Chequers. He agrees to her proposals, though it is later reported he had initial doubts. 9 July 2018 After vanishing for several hours, Johnson resigns.

James Duddridge, the MP for Rochford and Southend, made headlines on Wednesday morning by announcing that he had written a letter to Sir Graham Brady, the chair of the powerful backbench 1922 committee, demanding a vote of no confidence in the prime minister.

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If 48 such letters are submitted, Brady would be obliged to arrange a vote among the party’s MPs. If May lost, she would be barred from standing in the subsequent leadership contest – and cast out of Downing Street.

Yet Duddridge is perhaps best known in Westminster for attempting to lead a bid to depose John Bercow, the House of Commons Speaker, which ultimately gained little support.

Other MPs who have publicly claimed to have sent their own letters include inveterate critics of the leadership, Nadine Dorries and Andrew Bridgen.

But many others, even those who are deeply sceptical about May’s brand of Toryism, believe she deserves the right to pursue the Brexit negotiations to their conclusion – and some were encouraged by the one-nation vision set out in her speech.

Robert Halfon, the MP for Harlow, said he was encouraged that the prime minister appeared to have returned to some of the concerns she set out when she first arrived in Downing Street. “I liked it,” he said of her speech. “Not everything was there that I would like, but there was the signposting: on social justice, on cost of living, on housing. The proof of the pudding is in the eating, but I am encouraged.”

Other MPs simply fear that opening up the leadership question might yield an answer they like even less than May.

Scottish Conservatives feel so strongly about the need to avoid handing the reins to Johnson that they have reportedly launched a bid, nicknamed “Operation Arse”, to stop him.

Facebook Twitter Pinterest Scottish Conservatives have launched ‘Operation Arse’ to block Boris Johnson’s leadership aspirations. Photograph: Neil Hall/EPA

Asked about May’s leadership, friends and enemies single out the same quality – resilience. “She’s the great survivor,” said Anna Soubry, the former minister who has been a consistent critic of May’s approach to Brexit.

As negotiations with the EU27 reach their endgame in the coming weeks, that resilience will be sorely tested.

May and her team are braced for a gruelling set of discussions: with EU leaders, with her parliamentary partners in the DUP, and with the European Commission, all the while withstanding ferocious friendly fire.

Meanwhile, some of May’s cabinet colleagues believe it is vital to keep alive the plan B of a looser, Canada-style arrangement – in spite of the fact that it fails to solve the Irish border problem.

One of Johnson’s aims in launching such a ferocious public attack on the prime minister’s plans was to try to influence the decision-making inside cabinet; and some Brexiters insist May faces a greater threat from her colleagues around the table in Downing Street, than from restive backbenchers.

“There’s a general mood in cabinet of, we’ll give her one last chance to make progress; then we – to coin a phrase – take back control,” said one.

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Liam Fox, the trade secretary, offered his support to the prime minister’s approach on Friday – but only on the basis that the Brexit deal could be revisited later. “We should try to get as much of a final deal as we can get by 29 March, but it’s self-evident that if it’s a bilateral treaty, it can be revised later on,” he told Bloomberg.

Meanwhile, Philip Hammond’s task in drawing up his 29 October budget became significantly harder after the prime minister promised in her speech that a decade of austerity will end after Brexit.

Any spending increases would not be expected to kick in until next year’s spending review in any event; but it will be hard for the chancellor to avoid a shift of tone, after her upbeat rhetoric about voters’ sacrifices paying off.

Meanwhile, Brexit hardliners are prepared to resume the guerrilla parliamentary tactics they deployed against the government before the summer recess – potentially including, they claim, amending or even rejecting the budget, if it includes tax rises or other measures they don’t like.

But as to the question of whether the number of no-confidence letters from Tory MPs is about to hit the magic 48, Brady suggested at a fringe meeting in Birmingham that it was wise to be wary.

“The distance between what some of my colleagues say they might have done and what they actually have done can be considerable,” he said. “There are instances where I will see a Conservative colleague on the television saying they have written a letter to me when they haven’t. I will see them on the television saying they had withdrawn the letter to me, when they haven’t sent it in the first place. You have to be careful what you believe.”