Student organisations representing almost a million young people studying at UK universities and colleges are today joining forces to demand a referendum on any final Brexit deal, amid growing fears that leaving the EU will have a disastrous effect on their future prospects.

Predicting a young people’s revolt over the coming months, student unions – representing 980,000 students at 60 of the country’s leading universities and colleges – are writing to MPs in their areas this weekend, calling on them to back a “people’s vote” before a final Brexit deal can be implemented.

Student leaders said last night that they were planning action that would dwarf protests held in 2010 against the coalition government’s plans for student fees, and that they would not rest until they had been granted a say on their futures.

They argue in the letter to MPs that there are large numbers of young people – estimated at 1.4 million – who were too young to vote in the June 2016 EU referendum but who are now eligible to do so, and that this group deserves a say.

They also insist that promises made by the pro-Brexit groups during the campaign have not been kept and that only now, almost two years on from the narrow Leave vote, are most people beginning to understand what life outside the EU will look like.

“Because of all this, we call on our elected leaders to deliver on a people’s vote on the Brexit deal so that young people can once and for all have a say on their futures,” the letter says.

Among the university unions that have signed up are those representing students at Birmingham, Durham, Cambridge, Swansea, Leeds Beckett, Lancaster, St Andrews, Liverpool John Moore’s and Westminster. The joint letter and signatories were organised by the campaign group For our Future’s Sake (FFS).

Amatey Doku, deputy president of the National Union of Students, said: “When over 120 elected student officers, representing nearly a million young people, call for something with one clear voice, they need to be listened to. Students and young people overwhelmingly voted Remain and cannot see how the government can deliver a Brexit deal that works for them. As an elected representative body of 600 student unions, NUS is calling for a people’s vote on the Brexit deal.”

Both prime minister Theresa May and Labour leader Jeremy Corbyn have refused to back putting any final deal to the public, insisting that the British people have delivered their verdict and want to leave the EU.

But with May’s cabinet split over what form of customs arrangements the UK should propose for the post-Brexit period, and Corbyn facing an internal revolt by Labour MPs and peers – as well as signs of unrest over Brexit within the membership – an increasing number of MPs are coming round to the view that another national vote may be needed.

Last night, Melantha Chittenden, the national chair of Labour Students, which has a membership of 30,000, put Corbyn under further pressure, insisting that he had to listen to, and represent, their views. “Students want the Labour party’s policy to reflect their views and that means having a proper debate and vote on Brexit at the Labour party conference this year.

“It’s wrong to think students only care about student-specific issues like Erasmus – they care passionately about staying in the customs union and retaining freedom of movement, they understand the rights and protections that the EU affords us all and will do anything to defend that. That’s why young people voted Remain and it’s why we should get a say on the terms of the final deal.”

Meanwhile, there were growing signs that some supporters of the grassroots Labour campaign organisation Momentum will push for a shift in party policy on Brexit at the Labour conference in the autumn.

Michael Chessum, the former national treasurer of Momentum who is now national organiser of the leftwing anti-Brexit campaign group Another Europe is Possible, told the Observer: “The vast majority of Labour members – and the vast majority of Jeremy supporters – backed Remain in the referendum and haven’t changed their minds.”

Chessum said it was wrong for Corbyn supporters to argue that remaining a member of the EU would prevent the party implementing radical leftwing policies such as renationalisation. “The idea that EU rules will prevent us from enacting a radical programme in government is a fiction. There is nothing in our 2017 manifesto that could not be done, or is not already being done, within the EU – and a Corbyn government could lead the charge against state aid rules and liberalisation in the medium term. On the other hand, if we get into government to find a crashed economy, a bonfire of rights and regulations, and a series of trade deals which bind us to the American model, we’ll really struggle to fulfil our promises.” He said he expected “a big push” at the Labour conference for a referendum on the final deal.

New polling by Opinium, meanwhile, shows that Labour supporters are in favour of a people’s vote by 69% to 18%. The 18-34 age group support a people’s vote by a margin of 65% to 22%. Overall, 53% of the country supports the public having a vote on any final deal that the government agrees with the EU, compared to just 31% who oppose.

This week, cabinet ministers will make a fresh attempt to break the deadlock over the customs arrangements they will propose to the EU as part of the final deal. In a message that signalled her determination to press ahead with a hard Brexit the prime minister said last night that people should trust her to deliver on Brexit.

“I will ensure that we take back control of our borders. The public want their own government to decide on the number of people coming into Britain from across the European Union and that is what we are going to do,” May said. “I will ensure that we take back control of our money. We have agreed a settlement with the European Union and the days of vast contributions from taxpayers to the EU budget are coming to an end.

“So Brexit means there will be billions of pounds that we used to send to Brussels which we will now be able to spend on domestic priorities, including our National Health Service.”