Keir Starmer is under pressure to set out a timeline for his departure after a crushing defeat in elections across Britain prompted senior Labour MPs to call for him to step down within a year.

In a disastrous set of results, Labour lost hundreds of council seats in England, many to Nigel Farage’s Reform UK, which made big gains across the Midlands and the north as well as taking seats from the Tories in the South.

After more than a century of domination, Labour was also expecting a near-wipeout in Wales, where the party’s first minister, Eluned Morgan, lost her seat. Labour could slump to third place in Scotland behind the SNP and Reform.

While the prime minister appeared to have avoided an immediate coup, there was a furious response to the results among senior MPs and the unions, with some warning him to change course or risk electoral oblivion.

Louise Haigh, a former cabinet minister and co-chair of the powerful Tribune group of MPs, was the first to break cover.

“What is abundantly clear is that unless the government delivers significant and urgent change, then the prime minister cannot lead us into another election,” she said.

One senior backbencher said: “We want Keir to agree a timetable for his departure, but we want it to be dignified. He should have his last conference this autumn and then oversee a leadership contest straight after. He can’t take us into next year’s locals. It’s too late.”

However, Starmer insisted that he “won’t walk away” from the leadership as doing so would “plunge the party into chaos”, although he acknowledged that voters were fed up with the slow pace of change.

The prime minister would deliver a major speech next week in which he would seek to set out a more optimistic vision of the country’s future, Downing Street sources said.

The king’s speech on Wednesday will also include policy announcements, though a cabinet reshuffle is not expected.

On a tumultuous day:

• Nigel Farage said a “truly historic shift in British politics” had occurred after Reform UK won hundreds of seats and control of more councils in England. The gains included Essex where the Tory leader, Kemi Badenoch, has her constituency and which the Conservatives held for 25 years.

• Plaid Cymru became the largest party in Wales, beating Reform into second place, after Labour admitted it was on course to lose control of the Senedd for the first time since devolution. Morgan, the first woman to lead the Welsh government, became the highest-profile casualty and called on Labour to “go back to being the party of the working class”.

• The SNP leader, John Swinney, declared victory in the Holyrood elections – though was expected to fall short of an outright majority. The Scottish Labour leader, Anas Sarwar, conceded defeat saying his party had failed to counter “national dissatisfaction” with Starmer.

• The Greens gained their first two directly elected mayors – in Hackney and Lewisham – although they missed out on some more ambitious targets in London, as their leader, Zack Polanski, declared Britain’s two-party politics “dead and buried”. They also won three councils: Norwich, Hastings and Waltham Forest.

• The Tories were on course to lose hundreds of seats – both to Reform and the Liberal Democrats – across the south of England. However, they won back the flagship Westminster council in central London, with Badenoch announcing it meant the party was “coming back”.

• Labour appeared to be struggling in its London stronghold, despite early indications that its vote was holding up, unexpectedly losing control of Brent. Party insiders were closely watching councils including Lambeth, Lewisham and Haringey.

Despite Labour’s losses, allies of Angela Rayner and Andy Burnham, two of Starmer’s potential leadership rivals, said they would keep their counsel and watch how events developed over the weekend.

Wes Streeting was expected to speak to the media after his local council results. But they all appeared to have put any plans for a coup on hold.

As results were declared across the country, several other senior Labour MPs – many thought to be supporters of Burnham who usually refrained from openly criticising the prime minister – went public with their warnings.

Sarah Owen, the chair of the women and equalities committee, said: “Unless Keir Starmer delivers tangible change and truly connects with the public on a human level, he can’t lead us into another election – locally or nationally.”

Anneliese Midgley, the MP for Knowsley, said: “The results here and across the north are beyond our worst expectations. It’s truly devastating. Unless that changes significantly and quickly it’s clear the PM can’t lead us into another election.”

Simon Opher, the MP for Stroud, added: “The public has spoken; I think we need to change our leader. It’s not if now, it’s when. Keir is a decent guy but I think he’s not the right leader for the next election. If we go into the next election with him we will just get slaughtered.”

Cabinet ministers and Labour grandees rallied around Starmer as more backbenchers called for him to go.

The housing secretary, Steve Reed, said: “The British public don’t want to hear about timelines, backroom deals and navel-gazing. Let’s get on with the job.”

The business secretary, Peter Kyle, added: “Reversing these results requires a collective effort, not just blaming the boss. We can’t do that by turning in on ourselves. We do it by rebuilding faith in Labour’s ability to inspire and lead our country. Losing our PM is the very opposite of that.”

Several cabinet ministers have acknowledged privately that even with Starmer’s determination to stay in Downing Street after the election results, the mood on the backbenches is febrile and events could yet spiral out of control over the weekend.

They also admitted that – while they would discourage any move against Starmer now – they did not expect him to lead the party into the next election.

“When your personal brand is so poor, it is seldom retrievable,” one told the Guardian earlier this week.

Starmer also faced trouble in his own back yard, as Richard Olszewski, the Labour leader of Camden council, was one of three councillors who lost Holborn and Covent Garden ward to the Greens. He had switched from a seat in the north of the borough at risk from Lib Dems because this one was – supposedly – safer.