Labour MPs have been urged to stop plotting to remove Keir Starmer by Tom Watson, who as a junior minister spearheaded the last attempted coup against a Labour prime minister, when Tony Blair faced a revolt in 2006.

Watson’s warning came as Steve Reed, the housing and communities secretary, and a key Starmer loyalist, said Labour would risk “annihilation” if it decided to try to change leaders.

But with results for Labour expected to be particularly grim in Thursday’s elections for the Scottish and Welsh parliaments and English councils, senior party figures have told the Guardian that activists were being repeatedly told that the prime minister was the problem, rather than the party.

“They don’t hate Labour, they hate Keir, as unfair as that is, and I do think it is massively unfair,” one said.

There are nonetheless few expectations of a challenge soon after the elections, with expected challengers including Andy Burnham, Angela Rayner and Wes Streeting locked in what one cabinet minister called a “Mexican standoff”, with no one ready to move first.

Watson’s warning came in a Substack post in which he recounted his role in events in 2006, when Blair faced a letter from some MPs calling on him to set a date for his departure as prime minister, coupled with some junior ministerial resignations, Watson’s among them.

His advice to the current crop of Labour MPs, said Watson, who served as deputy leader under Jeremy Corbyn and is now a peer, was “not to be as reckless as we were in 2006”.

He went on: “Whatever the rights and wrongs of Labour’s current woes, the answer is not two-dozen backbench MPs writing a public letter calling on the prime minister to resign.” It would, he said, create a “Westminster psychodrama”, be seized on by opponents and go down extremely badly with voters.

“Voters will see a party talking to itself while the country is shouting at it,” he wrote. “The solution cannot simply be a different name on the door. The party has to listen harder, think deeper and recover its political purpose.”

Speaking earlier on Tuesday, Reed said he believed most fellow Labour MPs were not signed up to the idea of a challenge.

He told Times Radio: “The whole notion that we would copy the Conservatives and go doomscrolling through leaders in a way that means the government is completely incapable of dealing with the things that matter to most of the British public is absolute nonsense, and I’m not going to engage in it, and most of our MPs would not engage in that either.”

Steve Wright, the general secretary of the Fire Brigades Union, reiterated his call for Starmer to go in an interview published on Tuesday, saying that if results on Thursday were as bad as anticipated, the prime minister would be “a bit of a sitting duck”.

But one cabinet minister said an immediate challenge felt unlikely: “A number of candidates are in a Mexican standoff but nobody is ready to pull the trigger.

“It’s a reality that the next few months will be difficult. The idea that in that context the Labour party wants to turn in on itself and go round the country and have hustings, is frankly bizarre. My sense is that no one will trigger a contest. There will be a lot of noise and briefing and people will understandably be worried, but I think that Keir will still be here next week.”

Supporters of Burnham are thought to be waiting to see local election results across the north-west before potentially making an intervention on Friday evening. Backers of the Greater Manchester mayor say they will ask Starmer to set out a timetable for a dignified exit.

This timetable would give Burnham time to seek a seat in Westminster – and could mean he would no longer be obstructed by Labour’s national executive committee. If Starmer refused and Burnham sought to return to parliament nonetheless, he would almost certainly be blocked again.

Whatever happens, Burnham, Rayner and Lucy Powell, who replaced Rayner as Labour’s deputy leader, are expected to feel more free to express opinions once the election is over.

Some in Labour believe that such is the political hole Starmer faces, someone will act. One senior party source said: “Plenty of MPs now think they might as well just roll the dice and that anything would be better than where we are now.”