The Liberal Democrat surge has shown that Labour needs to get off the fence on Brexit and support remaining in the EU, the frontrunner to be the new Lib Dem leader has said.

Jo Swinson, who is leading the race to succeed Sir Vince Cable, said the Lib Dems had shown they could attract both former Labour and Tory voters fed up with their parties for failing to take a stand against Brexit.

She told the BBC’s Today programme: “Clearly Labour has had an atrocious night and absolutely needs to get off the Brexit fence. I spoke to so many former Labour voters, people in some cases who had voted Labour their entire lives, who voted Liberal Democrat for the first time on Thursday.

“We have really been bringing people together as the Liberal Democrats. To be able to get Michael Heseltine and Alastair Campbell united in the voting booth is no mean feat.”

In the results so far, the Lib Dems have finished in second place with 15 MEPs – more than the 11 MPs they have in the House of Commons and up from just one at the last European parliament election in 2014.

Cable, who is stepping down in July, said: “Our clear, honest, unambiguous message has won us our best ever European election result, and pushed Corbyn’s Labour into third place.

“We have shown ourselves to be the strongest remain force in British politics. We will always stand up for the people who have put their faith in us, taking this mandate forward to campaign harder than ever to stop Brexit.

“There is a clear lesson for Labour in tonight’s results. Get off the fence. In trying to please everybody they have pleased nobody.”

The party beat Labour to second place by more than 1m votes. It made significant inroads in Labour heartlands in particular, as voters abandoned it over its failure to unequivocally offer a second referendum.

Vince Cable (@vincecable) A clear, honest, unambiguous message has won @libdems our best ever European election result. We have shown ourselves as the strongest Remain force in British politics. Thank you to everyone who put their faith in us. We will stand up for you and keep campaigning to #StopBrexit.

Ed Davey, another potential successor to Cable, said the party was back in business and that the results overall showed that people wanted to halt Brexit.

He told the BBC: “The Liberal Democrats are the leading remain party. We have managed to garner a lot of votes from remain Conservatives and remain Labour and so I am very pleased the Liberal Democrats have had a fantastic night.”

He said it was legitimate to add up the Lib Dem vote with all the other votes for parties wanting to remain. “People want a chance to vote again. They want a final say,” he said.

Facebook Twitter Pinterest The Lib Dem MP for Kingston and Surbiton, Ed Davey. Photograph: Isabel Infantes/AFP/Getty

The party’s resurgence was exemplified by its performance in London, where it went from zero MEPs to three. In the Labour stronghold of Islington, where Corbyn is MP, the Lib Dems’ share of the vote was up 18.6 points and Labour’s down 21.2. It was a similar story in neighbouring Haringey, where it was up 20.2 points and Labour down 17.7, placing the Lib Dems first.

The Lib Dem vote was up 30 points in Kingston upon Thames and a similar amount in Kensington and Chelsea, both areas where the Tory share was down by over 20 points. The party also swept Camden, Lewisham and Sutton, helping to make it the largest in London, the only region in England and Wales where that distinction did not go to the Brexit party. The overall Lib Dem vote in the capital was up 20 points.

The party also made telling gains outside the capital, in some firmly remain areas but also acting as a counterweight to the Brexit party in leave-voting areas. In the East of England, where Ukip and the Tories shared the six seats equally between them at the last European election, the Lib Dems won two and the Greens also picked up one, with the Brexit party taking the remainder. The Lib Dems won two seats to the Brexit party’s three in South West England and one in Yorkshire and Humber and the West Midlands, both areas where the Brexit party won the largest share of the vote.

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It was not just the general public who deserted the two main parties for the Lib Dems. Campbell, the people’s vote campaigner and former spin doctor to Tony Blair, revealed that he had not cast a vote for Labour and instead voted for the party pledging to try to stop Brexit.

EU elections: Tories and Labour savaged as voters take Brexit revenge Read more

“I didn’t vote Labour for the first time in my life and it was a very, very strange feeling,” he said. “But I just felt on this issue, at this time, the Labour party has let its own supporters down, it has let its own members down and I think it has let the country down in the way that it has failed properly to devise a policy that the country and the party could unite around, and the way that it failed to campaign.”

Alastair PEOPLE’S VOTE Campbell (@campbellclaret) I confirmed to @lbc I voted Lib Dem, as did @schooltruth - for both of us the first time ever we did not vote @Uklabour - I’m not a Lib Dem. I’m Labour and I hope that in voting as I did I will help the Labour Party see sense and do right thing for the country @peoplesvote_uk

Heseltine, who had the Tory whip suspended after saying he would vote for the Lib Dems, confirmed that he had gone ahead with his threat. “I did what I believed to be in the national interest, and it fulfilled a warning that I gave to my party many months ago that the Lib Dems would take a significant number of Conservative votes, which they have done.”