‘You want this Brexit process to be over and done with.” Last Wednesday, Theresa May presumed to speak for the whole nation in a catastrophically judged TV address to the public. Yet, as Britain teeters on the edge of crashing out of the EU without a deal, which would surely be the worst national disaster since the Second World War, the disconnect between voters and the actions being taken in their name has perhaps never been greater. Just 14% of voters say they think the deal the prime minister is trying to ram through parliament delivers the type of Brexit people voted for in 2016.

It is this yawning gulf between public sentiment and government action that propelled more than a million people from all over the country to take to the streets of the capital today in a compelling demonstration of disdain for the prime minister’s tactics and support for a popular vote on the terms of Britain’s exit from the EU. Critics have accused the protesters of staging a last-ditch attempt to overturn the result of the referendum. But the historic march represents something far more significant: the growing number of voters unwilling to allow politicians in thrall to ideological obsessions or narrow party interests to use an ill-defined and chimerical notion of the “will of the people” to drive through divisive and potentially catastrophic agendas.

Theresa May has consistently sowed division where she should have tried to generate national consensus

The marchers have made their voices heard. They must be heeded. The initiative now passes to parliament. Will MPs heed the reminder that more than 16 million people voted to remain in the EU in 2016 and that the most important job of government in the wake of a narrow victory for Leave was to try to forge a path that healed a divided country?

May has consistently sowed division where she should have tried to generate national consensus. She has pursued a Brexit strategy designed first and foremost to keep the Conservative party unified and her premiership secure, the national interest barely an afterthought. And she has ridden roughshod over parliament at every turn: resisting giving it a say over triggering article 50; granting a meaningful vote on her deal only when compelled to do so; attempting to force MPs into voting for her deal by delaying vote after vote to take us closer to the cliff edge.

May’s televised address last week was the culmination of that strategy. “I am on your side,” she intoned to voters. “Parliament has done everything possible to avoid making a choice.” She embraced populist language that could have been uttered by any tinpot dictator looking to trample the democratic institutions frustrating their personal agenda. In our parliamentary democracy, May’s mandate to lead the country comes purely from any support she commands from the House of Commons. Her words were not only self-defeating but bordering on the dangerously unconstitutional.

The 2016 referendum established nothing more than that there was a narrow majority of voters in favour of leaving the EU. No concrete plan for leaving the EU was presented during the campaign; on the contrary, it was characterised by contradictory claims and a denial that Brexit would involve trade-offs. Yet May invented a mandate that would bolster Tory unity, adopting red lines on scrapping freedom of movement and leaving the customs union that reflected her party’s Eurosceptic flank rather than public sentiment. In a citizens’ assembly, run by University College London’s constitution unit in 2017, whose members were demographically representative of the UK and of the Leave-Remain split, 70% were in favour of maintaining freedom of movement by the end of the deliberative process, so long as the UK made full use of existing controls. The real impetus for May’s Brexit comes not from the “will of the people” but from the hard right of the Conservative party.

While the march was taking place, Jeremy Corbyn was canvassing in Morecambe. Photograph: Jason Roberts/PA

Labour’s position on Brexit has been no less parochial. Jeremy Corbyn has sought to maintain an ambiguous position on Brexit for as long as possible, ignoring the views of a huge proportion of his party members. He has failed to capitalise on May’s weaknesses to provide the leadership Britain so desperately needs. Both parties are riven by Brexit divides, rendering the political system helpless to respond to the crisis engulfing Westminster.

Even though parliament had expressed its opposition to leaving on Friday without a deal, this looked distinctly possible as a result of the dangerous political stasis that had set in. But at the 11th hour, the EU granted a reprieve, postponing the Leave date by two weeks to 12 April, the cutoff for the UK notifying the EU it will take part in this year’s European elections. May has until then to get her deal through. If not, we must either request a longer extension, contingent on taking part in European elections, or leave without a deal.

The calls for May to resign from within her own party have increased in intensity this weekend. But this would achieve nothing save plunging the Conservative party into a leadership contest and the country into even deeper crisis. The only way forward is for parliament to wrest control from the government this week and hold a series of indicative votes to try to establish which route commands most support in the Commons.

The only way forward is for parliament to wrest control from the government and hold a series of indicative votes

Several MPs have been advocating a Norway-plus type deal. This is certainly the least bad form of Brexit, although it would significantly reduce the UK’s influence over the laws and rules with which it would need to continue to comply. But a major question mark hanging over this proposal is whether the cross-party coalition in favour of Norway-plus could be sustained over the months, even years, that it would take to negotiate. There will be no agreement about the UK’s future relationship until after Brexit. A soft-Brexit coalition of MPs could risk handing a blank cheque to any future prime minister to negotiate whatever agreement they see fit and present this versus no deal to MPs at the end of the transition phase.

The best option by far remains a referendum on May’s withdrawal deal versus the status quo. The 2016 referendum put no concrete proposal to leave the EU on the table and it would be unthinkable to leave without asking voters to ratify the terms of our exit. If voters choose to support May’s deal, Brexit must be followed by a national conversation about the UK’s future relationship with the EU, informed by a national citizens’ assembly.

May has shown, time and again, that she is on the side of whatever she believes best binds the Conservative party together. It is entirely conceivable that, in the event of her deal being voted down again, she ignores parliament and opts for no deal to secure her legacy as the prime minister who delivered Brexit. On Saturday, Britons turned out in their many hundreds of thousands to deliver their judgment of the strategy of May’s government so far. Now the only people who can stop her are our elected representatives. Parliament has never faced a more important test.