Theresa May and Jeremy Corbyn have vowed to press ahead with seeking a cross-party solution to the Brexit deadlock at Westminster, after voters punished both major parties in local elections.

The Conservatives’ net loss of more than 1,300 seats on their 2015 figures marked their biggest defeat since John Major was prime minister. Disillusioned voters deserted the party in droves, including in traditional Tory areas such as Chelmsford and Surrey Heath.

Labour had expected to make gains, but instead suffered a net loss, and lost control of a string of councils, including Burnley, Darlington and Wirral.

Vince Cable’s remain-supporting Liberal Democrats were the major beneficiaries, taking control of 10 councils, including Cotswold and Winchester, while the Greens and a string of independents also fared unexpectedly well.

The Greens’ co-leader Sian Berry described her party’s performance as “a spectacular 24 hours”, adding: “Voters are clearly fed up with the tired old politics of the parties of the past, who have delivered a UK in Brexit turmoil.”

Cable told celebrating Lib Dem activists that the result reflected a “story across the country”. He said: “The Lib Dems were written off at one point, but we’re coming back very, very strongly. We’re the big winners of the night throughout the country.”

Many Labour MPs suggested the results underlined the urgency for Labour to shift to a full-throated remain position.

But Corbyn insisted: “I think it means there’s a huge impetus on every MP, and they’ve all got that message, whether they themselves are leave or remain – or the people across the country – that an arrangement has to be made, a deal has to be done. Parliament has to resolve this issue – I think that is very, very clear.”

Close Corbyn allies Ian Lavery and Richard Burgon echoed his message, saying Brexit was detracting from a string of other crucial issues, while shadow chancellor John McDonnell said the message from voters was: “Brexit – sort it.”

The prime minister, who was heckled by a party activist as she began to address the Welsh Conservative conference, said the voters were giving a “simple message” to the Conservatives and Labour: “Just get on and deliver Brexit.”

She conceded that the results were “very difficult” and apologised to councillors who had lost their seats, saying they were not to blame.

Scottish Conservative leader Ruth Davidson, speaking at her party’s conference in Aberdeen, said: “I think the message is pretty clear. It seems to be ‘A plague on both your houses.’”

The projected national share of the vote, calculated by elections analyst John Curtice for the BBC, put both major parties neck-and-neck on 28% of the vote – both down from 35% a year ago. If that result were replicated in a general election, it would result in another hung parliament.

Curtice said the results demonstrated that Labour had been hit just as hard by the Brexit logjam over the past 12 months as the deeply divided Tory party. “The opposition is in no way demonstrating an ability to profit from the government’s misfortunes,” he said.

Labour MPs who favour a referendum said the results showed that Corbyn’s attempt to appeal to both remain and leave voters had failed, and he should throw his weight behind a people’s vote.

Sunderland MP Bridget Phillipson said: “I fear Labour’s position has been too hesitant and lacking in clarity over the past few months, depressing support among our voters at a time when they expect strength and leadership from my party rather than fudge.”

However, Corbyn appeared determined to press ahead with seeking a compromise with the Conservatives – a move that could spark a ferocious backlash from many of his MPs unless it is accompanied by the promise of a “confirmatory” referendum.

Participants on both sides of the cross-party Brexit negotiations reported a marked improvement in tone at the start of this week, and several senior Labour figures, including the shadow business secretary, Rebecca Long-Bailey, have played up the idea of a deal.

Both parties fear the disappointing set of council results could be dwarfed by the challenge facing them at European parliament elections in three weeks’ time, when political newcomers Change UK and the Brexit party will be standing candidates.

The foreign secretary, Jeremy Hunt, suggested he was optimistic about the prospects of a deal, and hoped that Labour might sign up to some compromise that brought “the benefits of a customs union” but would also allow the UK to have an independent trade policy for the services sector.

Hunt suggested that the local election results, and trends in the polls, gave an incentive to both parties to compromise since both parties were being dragged down. He added: “It is actually in both parties’ interests to resolve this because we will both be punished equally hard by our core voters.”

Hunt, a former remainer who is now willing to countenance a no-deal Brexit, warned that the electorate had become angry and bored. “They want it resolved. They want to move on, so anyone thinking of delaying Brexit further needs to remember one of the core reasons why the people of Britain decided in large numbers to give a punch in the face to the British establishment – which is overwhelmingly remain – is because they thought we were not listening to them.”

“If, three years on from that vote, we still fail to deliver Brexit, then that anger will only amplify.”

• This article was amended 6 May 2019 because an earlier version described Rebecca Long-Bailey as shadow Brexit secretary instead of shadow business secretary.