Prime Minister Theresa May was confronted with anger from her own party after local election results which she admitted were “very difficult” for the Conservatives.

The Tory leader oveersaw a local election massacre, with one-in-four of her councillors being booted out of their seats.

Backbench MPs called for her removal and warned that the party would be “toast” if it did not change direction, and a heckler interrupted the PM as she gave a speech in Wales, saying: “Why don’t you resign? We don’t want you.”

Voters across the country vented frustrations over Brexit on both major parties, with the Liberal Democrats, Greens and independents benefiting from a hammering for both the Tories and Labour.

Lib Dem leader Vince Cable described the party’s local elections performance as “brilliant”.

The Twickenham MP told BBC Breakfast they were “the best results we’ve had in the 40 years of our existence”, adding that the party’s opposition to Brexit will help them in the upcoming European elections.

“When people are trying to make their minds up, they would and they should vote for us, knowing that every vote is a vote to stop Brexit.”

But Labour MP Lisa Nandy said that Brexit was “enormously overstated” as a factor, blaming poor results for the two major parties on “people losing faith with the system as a whole”.

Labour’s chances were harmed by “the fact that people aren’t clear what our policy is” on Brexit, she said.

She added: “The message we’ve got to take to the country is actually that we’ve heard people loud and clear, that the good jobs have left areas like mine, young people have had to move away, we don’t have spending power any more, the high streets are collapsing, the bus networks have gone.

“The only way for Labour to win the next general election is to rebuild that coalition that has brought us into power on three occasions in history.”

With all results from local elections in England declared, the Conservatives lost more than 1,200 seats – their biggest defeat since John Major was prime minister.

Councils including Chelmsford, Winchester and Bath fell directly into the hands of the Lib Dems.

But Labour was also licking its wounds after forfeiting control in heartland councils like Burnley, Hartlepool and Bolsover.

Despite some predictions that Jeremy Corbyn’s party could pick up three-figure gains, Labour was down more than 100 seats, though it did have the consolation of restoring control in Trafford for the first time sine 2003.

Both the Conservative and Labour high commands left no doubt that they saw the results as a demand for resolution of the Brexit impasse three years after the 2016 vote for EU withdrawal.

May welcomed the Labour leader’s offer to get a Brexit deal done as the only escape route, during a speech in Grimsby, one of the few areas in which the Conservatives enjoyed success.

She said: “I welcome the fact that Jeremy Corbyn has said today that he sees the time is now to get a deal and to deliver on Brexit – it’s what I’ve been saying for some time.

“It’s what we want to do, it’s what we’ve been working for, so now we must get on and do that.”

May told the Welsh Conservatives in Llangollen: “I think there was a simple message from yesterday’s elections to both us and the Labour Party – ‘Just get on and deliver Brexit’.”

Health Secretary Matt Hancock said that the message from voters in local elections was: “Get on, deliver Brexit and then move on.”

He indicated that he was prepared to back compromise with Labour’s call for a post-Brexit customs union arrangement with the EU.

“I think the Prime Minister’s deal is a better arrangement than a permanent customs union, but I think we need to be in a mood for compromise,” he said.

“We need to be listening to these results from these local elections which are about ‘deliver Brexit’, not ‘deliver this particular form of Brexit’.”

He suggested both sides in cross-party talks would have to shift on their Brexit stance: “I think we do need a mood for compromise, but compromise often involves looking at the different positions of different groups and coming up with something in-between.”

Remain-backing Labour MPs warned the leadership against striking a Brexit deal without the promise of a referendum, after shadow cabinet minister Barry Gardiner suggested the party was “bailing out” Tories in cross-party talks.

Ilford North MP Wes Streeting said: “Labour should not be bailing the Tories out. Any deal – any – must go to a public vote. Without a commitment to a public vote, I’ll vote for a Labour-Tory deal when hell freezes over and I’m not alone in that.”

There were calls from Tory MPs for May’s removal as leader, with senior Brexiteer Sir Bernard Jenkin warning that the party would be “toast” unless it “mends its ways pretty quickly”.

He said voters overwhelmingly believed that the Prime Minister had “lost the plot” and that the time had come for a change of leader.

“Certainly among Conservative activists and council candidates there is an almost universal feeling that it is time for her to move on,” he said.

His comments were echoed by former Cabinet minister Priti Patel, who said voters saw May as “part of the problem”.

“I just don’t think we can continue like this. We need change, we need a change of leadership. Perhaps the time has now come for that,” she told the BBC.

With all results in the Conservatives had lost 1,269 seats, Labour 63 and Ukip 36.

The Lib Dems gained 676, the Greens were up 185 and independents increased by 242.

Conservative Party chairman Brandon Lewis blamed the failure to resolve Brexit.

“People are frustrated with where they see parliamentarians are and the fact that we have found this impasse in Parliament,” he said.

“It’s a stark reminder to everybody in the House of Commons that we need to get past that impasse, deliver on what people voted for, and focus on that as parliamentarians as well.”

The Conservatives lost councils including Peterborough, Basildon, Southend, Warwick and Worcester to no overall control, while Winchester, Chelmsford, Bath and North East Somerset, Somerset West and Taunton, Mole Valley, Vale of White Horse, Cotswold and Hinckley and Bosworth fell to the Liberal Democrats, with North Kesteven going to independents.

However the party held on in the bellwether council of Swindon, seen as a possible Labour gain, and took Walsall and North East Lincolnshire from no overall control.

Labour, meanwhile, lost control in Darlington, Middlesbrough, Stockton and Wirral and the mayoralty in Middlesbrough, where its vote was down 11% as independent Andy Preston was elected, although it did gain control of Amber Valley from Tories.

Even where the party held on in its traditional stronghold of Sunderland, which voted heavily for Brexit in the 2016 referendum, it still lost 10 council seats.

Council leader Graham Miller said the party had paid the price for its stance on Brexit, with some MPs calling for a second referendum.

“The people of Sunderland have said ‘We are just not accepting that’. We have seen a massive protest vote on that issue tonight,” he said.

Visiting Trafford to congratulate local activists, Mr Corbyn said the results were “interesting, to put it mildly”.

“I wanted us to do better, of course,” he said. “But I also say the swings to Labour in many parts of the country show that we can win seats in a general election, whenever that comes.”

Polling expert Professor Sir John Curtice, of Strathclyde University, said the voters appeared to be punishing whichever of the main two parties was in control in their area.

“The Labour Party is losing where they are strong historically, the Conservatives are losing where they are strong historically. It’s a plague on all your houses,” he said.

In contrast, the Liberal Democrats, who fought on a pro-Remain platform, were in buoyant mood.

As well as picking up councils from the Tories, they took North Norfolk, Teignbridge and North Devon from no overall control.

Home affairs spokesman Sir Edward Davey said the results were “equivalent to our best strides forward ever in our history”.

“We are clearly back in the game,” he said. “The Liberal Democrats have proven we are that strong alternative to the Tories and Labour.”

Green Party co-leader Jonathan Bartley said: “This is the biggest election night in our history. Greens are winning right across the country, and taking seats from a wide range of other parties.

“The Green message is clearly taking hold and can win anywhere.”