Like a desiccated woman who is crawling across a scorching desert and thinks she has spotted a source of moisture, for a brief moment Theresa May tasted the sweet tang of relief from her miseries.

She had about 72 hours of experiencing what it feels like to be a prime minister who is in control. This time last week, she had launched her Brexit plan and it had not instantly exploded on takeoff. EU leaders gave the blueprint a polite, if highly guarded, reception. No one flounced out of the Chequers meeting, saying they could no longer sit in her cabinet. Even critics expressed some backhanded praise for the way in which Mrs May choreographed the corralling of her ministers. Opponents revived the old charge, an accusation not heard against her since she blew last year’s election, that an imperious leader had railroaded ministers into signing up to the plan. The poor things had been denied their mobile phones and told that it was a long walk home if they were thinking of quitting.

This narrative of a mission triumphantly accomplished survived for a short period even when David Davis resigned and was followed, some 18 hours later, by Boris Johnson, whose departure from the Foreign Office was as chaotic, self-indulgent and self-serving as his time in the post. Mrs May was not entirely surprised. Threats to resign by Mr Davis had become an almost weekly event; she would have been happy to see the back of Mr Johnson long ago. Her aides had wargamed likely resignations and thought about replacements.

The outgoing Brexit secretary drew some of the sting from his resignation by declaring that he had not quit with the intention of trying to bring down the prime minister. The departure of the foreign secretary was openly celebrated by his own officials and unlamented by everyone else. By getting other cabinet Brexiters, notably Michael Gove and Andrea Leadsom, to publicly own her plan, the Tory leader appeared to have pulled off a tactical coup by splitting the Brexiters between Compromisers and Quitters.

Yet for all that superficially successful politicking, her Chequers gambit has not stood the test of time. Barely more than a week later, it is already turning to dust. And her premiership is back in “turmoil”, to quote one of the few trustworthy things said by Donald Trump.

He wasn’t the central cause of her renewed miseries, but the American visitor certainly deepened them by displaying a viciously blatant contempt for his host. He savaged her plan in remarks to Britain’s biggest-selling newspaper, which were published just as he was having dinner with the prime minister. His subsequent attempts to bind up some of the lacerations he inflicted on his host only drew attention to how severe the wounds were. His suggestion that Mrs May had betrayed the Brexit vote and “killed” any prospect of a trade deal with the United States hurt her in two ways. This supplied ammunition to the fundamentalists in her party. At the same time, the Trumpian rampage underlined for Remainers what folly it is for Britain to be separating itself from Europe just at the moment when there is a rogue president in White House.

He made things worse, but neither he nor the resignations from the cabinet are the fundamental problem for Mrs May. It is the maths. People have started to do the parliamentary maths. Lyndon Johnson, who was majority leader in the US Senate before he became his country’s president, once declared that the most important talent in politics is “the ability to count”. There aren’t enough people who can count around Mrs May. The fatal flaw in her plan is that there is no majority for it in the House of Commons.

‘Jacob Rees-Mogg and his cabal can muster the 48 signatures of Tory MPs that they need to trigger a confidence vote in Mrs May.’ Photograph: Simon Dawson/Reuters

The Brexit ultras are crying treachery and promising havoc. They both express and feed the furies of Tory activists. The Brextremists don’t have an alternative plan, other than to crash out of the EU without any deal at all, a catastrophic outcome that some of them actually wish for, but that hasn’t stopped them before and won’t curb them now. Jacob Rees-Mogg and his cabal can muster the 48 signatures of Tory MPs that they need to trigger a confidence vote in Mrs May. They do not sound confident that they have the numbers – they require 159 – to oust her from the premiership. What the ultras can do is make the government’s life even more hellish by prosecuting a “guerrilla war” in parliament. Even if Mrs May could get the EU to accept her plan, 60-plus Conservative MPs are opponents of her version of a Brexit deal. That number will climb if, as is inevitable, she has to make further concessions in Brussels to secure an agreement. There are more than enough Brextremist rebels to block the prime minister in the Commons unless she can get some assistance from the opposition.

She needs the help of Labour MPs and she is not going to get it. Jeremy Corbyn won’t give her any succour. He is more interested in bringing down the Tories than helping them to solve a mad riddle of their own making. The Labour leadership calculates that defeating Mrs May in Brexit votes is their best chance of collapsing the government and precipitating an early general election. But Number 10 clearly harboured hopes that centrist Labour MPs might embrace her plan as the least worst version of Brexit that they are likely to get in the circumstances. David Lidington, the minister for the Cabinet Office, was sent out to try to woo Labour MPs. This succeeded in giving the Moggites something else to foam about, but failed to recruit support from the opposition. “It didn’t work,” says one Labour parliamentarian. “Labour MPs went along [to hear Lidington] to find out what the Tories were up to – not to buy into the plan.” It is a big ask at any time to get opposition MPs to rescue a prime minister from her own party. In the current climate, no Labour MP fancies going back to their constituency to explain to party activists why they helped save the neck of a Tory prime minister. Even more importantly, Labour MPs aren’t going to throw a lifeline to Mrs May because they think her plan is no good. They may accept that it does represent some extremely belated recognition of the damage that will done to the economy by a bad Brexit. They will also acknowledge that the prime minister has inched somewhat closer towards a sensible form of departure from the EU. But a plan that is too soft a version of Brexit for the ultras is still too economically perilous to secure the support of opposition MPs.

All this before her proposal has even made serious contact with the negotiators for the EU. The initial response from European capitals was muted because it would have looked mean-spirited to instantly throw the plan back in her face. That is very different to saying that European leaders are just going to accept the ideas presented by the prime minister. They are too polite to say that her white paper is rubbish, but there are key parts that they regard as unworkable or unacceptable.

Two years have elapsed since the Brexit vote and only now has the government produced a negotiating position that the EU can engage with. In that wasted time, the atmosphere of the talks has become terribly sour. Even if the EU is prepared to entertain Mrs May’s proposed split in the single market between goods and services – and that’s one of several very large ifs – reaching a deal is bound to involve more concessions in Brussels and to the economic realities of Brexit.

So Mrs May’s grand plan has left her stranded in no woman’s land. She can’t go back. Abandoning her plan and retreating behind her old red lines might win a temporary respite from the furies of the Brexit ultras, but that relief would be purchased at the cost of shredding what remains of her authority over the government and destroying what’s left of her credibility in Europe. Mrs May can’t stand still, not for very long anyway, because the EU won’t buy her plan as it is and she hasn’t got a majority for it in parliament. She can’t go back. She can’t stand still. She can only go forward. If, that is, she has any strength left to go forward. That would involve taking further steps towards a softer version of Brexit, a move that would arouse even more intense rage against her from within her party.

The oasis in the desert was a mirage. There is no relief on the horizon from Mrs May’s agonies. The desiccated woman is still sucking up hot sand.

• Andrew Rawnsley is an Observer columnist