David Cameron is under growing pressure to soften his hardline deficit reduction strategy after a wave of redundancies in central and local government sent unemployment surging beyond 2.5m.

With the City predicting joblessness would hit 2.75m next year, the Institute of Directors, the Prince's Trust and the TUC joined the opposition in demanding urgent action to boost the flagging economy.

Cameron admitted the official figures – which included the highest female unemployment in 23 years and almost a million young people shut out of the labour market – were "disappointing".

But he insisted that the coalition would not do a U-turn as it attempted to repair Britain's public finances over the course of the current parliament. He said: "All governments are having to take difficult decisions about cutting public spending. Anyone standing here would have to make those decisions. This government is reducing the welfare bill and reforming public sector pensions. If we weren't taking those steps you would have to make deeper cuts in the rest of the public sector."

Ed Miliband, the Labour leader, said the government's plan for an expanding private sector to replace jobs lost as a result of the austerity programme was not working after the Office for National Statistics reported on Tuesday that 111,000 jobs were lost in the public sector in the three months to June 2011, against 41,000 created in the rest of the economy.

"The message to all those people who have lost their jobs is the prime minister is not going to change course," he said. "For every two jobs being cut in the public sector, less than one is being created in the private sector. Isn't that the clearest sign yet that your policy just isn't working?"

Ministers had been preparing for poor unemployment figures after evidence emerged in recent months to show the economy's recovery from the deep recession of 2008-09 had almost stalled.

The ONS said joblessness was rising on both measures used by the government, the internationally agreed Labour Force Survey (LFS) and the more narrowly based claimant count.

Using the LFS yardstick, unemployment stood at 7.9% in the three months to July, while a 20,300 jump to 1.58m in August left the claimant count jobless rate at 4.9%. Unemployment among the under-25s rose by 77,000 in the three months to July, taking the total of unemployed 16-24 year olds to 972,000.

A spokeswoman from The Prince's Trust youth charity said: "It is deeply concerning that youth unemployment has risen sharply, with young people hit hardest and those out of work for more than a year increasing by nearly a fifth. To tackle this downward spiral of youth unemployment, government, businesses and charities need to work together on schemes that work. More than three in four young people supported by The Prince's Trust last year moved into work, education or training."

Scotland was the one region of the UK to see a fall in unemployment between May and July. Alex Salmond, the first minister, said the decline was due to extra spending on infrastructure projects and support for small and medium sized companies.

Analysts said the weakness of the labour market was highlighted by a fall in vacancies, a 40% jump in redundancies and a record number of people working part-time but in search of full-time jobs.

Scott Corfe, senior economist at the Centre for Economic and Business Research, said: "The UK government will now be under immense pressure to deal with unemployment, especially given President Obama's announcement of a $450bn (£280bn) job creation package in the US last week. The focus of a UK jobs creation package would almost certainly be on private sector deregulation and measures aimed at reducing the risk associated with hiring new workers – rather than a slowdown in the pace of deficit reduction – given the political costs of moving away from plan A."

Graeme Leach, chief economist at the IoD, said it was time for the Bank of England to announce a second round of quantitative easing (QE), with the Bank of England buying bonds in order to create money. He said: "The storm clouds are gathering, with falling employment and rising unemployment at a time when it is difficult to see how this might reverse. Today's figures reinforce our belief that we need to launch QE2 as soon as possible."

The TUC general secretary, Brendan Barber, said: "These are terrible figures. They are further evidence that the recovery has been choked off by a self-defeating rush to austerity. Government policies are hurting, but they aren't working."

Nigel Meager, Director of the Institute for Employment Studies, said: ""It's hard to avoid the conclusion that policy-makers now need to stop sitting on their hands and start looking for ways to get spending power into the economy quickly."