Theresa May is accused today of breaking her promise to champion the life chances of people from poorer backgrounds, after the failure to appoint a chief adviser on the subject for a staggering four months.

The appointment of a new head of the Social Mobility and Child Poverty Commission has been delayed, The Independent has learned, prompting a claim the government wants someone who will not make embarrassing criticisms.

It means the entire commission is still in paralysis – exactly four months after Alan Milburn and his team quit in despair at the lack of policies to create “a fairer Britain”.

Ministers fear appointing an ally who will be seen as part of a “chumocracy”, particularly after the fiasco when Toby Young was picked to head up a new universities’ watchdog.

However, they are also believed to be reluctant to pick someone who will repeat Mr Milburn’s regular, hard-hitting censures of the government’s social mobility record.

“They are looking for someone more aligned to the government’s agenda and are wary of the more academic types whose behaviour is harder to predict,” said one ex-Tory minister familiar with the process.

And another insider said: “It’s a fair point that it’s been slow. No one wants it to look like a chumocracy, so it can’t be a Tory.”

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It is understood that the selection process was due to conclude last month, but the hunt for a new chairperson is continuing.

David Lammy, a Labour MP who has campaigned on social mobility, seized on the delay, saying: “This frankly says everything about how seriously this government takes social mobility – lots of rhetoric, no action.”

And Layla Moran MP, the Liberal Democrat education spokesperson, said: “Alan Milburn was a former Labour minister appointed by a Liberal Democrat deputy prime minister.

“This represented a true cross-party effort to make our society more fair. This all seems long ago as the Conservatives unpick much of the good work done throughout the coalition government.

“May must refill this post so that we can start to deal with problems in our society including social mobility and intergenerational fairness.”

However, the aforementioned source said the government recognised it had no option but to keep the commission going, adding: “It was set up by legislation – there is a legal duty to maintain progress on social mobility.”

The prime minister was plunged into a crisis when Mr Milburn and three of his fellow commissioners – including a leading Conservative, Gillian Shephard – walked out in December.

The former Labour cabinet minister said dealing with Brexit had left the government without “the necessary bandwidth to ensure the rhetoric of healing social division is matched with the reality”.

“I have little hope of the current government making the progress I believe is necessary to bring about a fairer Britain,” he wrote in a stinging letter to the prime minister.

He added in an interview: “The worst position in politics is to set out a proposition that you’re going to heal social divisions and then do nothing about it. It’s almost better never to say that you’ll do anything about it.”

Mr Milburn protested that the number of commissioners had shrunk from 10 to four, as key posts went unfilled for years, and hinted at rows over its remit and resources.

The resignations came hard on the heels of a report warning of a “striking geographical divide”, leaving many parts of the country behind as London and the South-east pulled away.

They were hugely embarrassing for the prime minister, who entered Downing Street promising to make life easier for people in “an ordinary working-class family” and those “just about managing”.

In a major speech at the start of 2017, she pledged to tackle “everyday injustices” afflicting “those who feel the system is stacked against them”.

“People who are just managing, just getting by, don’t need a government that will get out of the way, they need a government that will make the system work for them,” Ms May said.

“An active government that will allow them to share in the growing prosperity of post-Brexit Britain.”

The Social Mobility and Child Poverty Commission was set up in 2010, in the dying days of the Labour government, charged with carrying out research and producing an annual report.

Its responsibilities also include “providing advice to ministers (at their request) on how to improve social mobility in England”, which “must then be published”.