The Government’s social mobility adviser will urge Theresa May to replace David Cameron’s flagship “big society” scheme for 16 and 17 year-olds with high quality work experience and careers advice for all teenagers.

Alan Milburn, who chairs the Social Mobility Commission, wants the £400m-a-year budget for National Citizen Service (NCS), which includes summer camps and community projects, switched to boosting the job prospects of all young people.

His annual “state of the nation” report, to be published on Wednesday, will propose that the £1,500-per-head spent on those doing NCS be weighted towards children from disadvantaged families in future. It will warn that they often miss out because they go to the worst schools, offering little or no careers advice or work experience, even though they need the most help. In contrast, children from better off families are more likely to get guidance and work placements because they attend the best schools and their parents know how to play the system.

Critics claim Mr Cameron’s flagship project offers poor value for money and has not provided enough opportunities for children from poor families. It has been dubbed a “CV-enhancing service” for school-leavers going to university or into jobs. Its budget is due to rise from £100m in 2015-16 to £400m by 2019-20.

Mr Milburn’s proposed shake-up could appeal to Ms May, who has ditched some of Mr Cameron’s policies since succeeding him in July and promised to champion working class families rather than the “privileged few.”

Mr Milburn will make improving skills to meet the challenge of the post-Brexit era a key theme of his annual report. He was alarmed to discover that fewer than half of 16-year-olds have done any work experience.

The former Labour Cabinet minister will argue that good careers advice can add about £2,000 a year to future earnings. Despite that, there is no guarantee children receive such guidance, which is left to the discretion of schools, no penalty for schools not providing it and no money ring-fenced for it by the Government.

He will cite evidence that pupils from better off families are one and a half times more likely to get careers advice than the less advantaged, who are less satisfied with such guidance when they do receive it.

Mr Milburn will warn: “For too long, those that make good choices in the education system have been the children of parents who are familiar with the system, having successfully navigated it themselves. Ensuring that young people from all backgrounds are prepared for the world of work is an important feature of any education system that seeks to ensure that all children can fulfil their potential.”

He wants NCS to be transformed into an “opportunity fund” so that all 14-18 year-olds can receive a week-long work experience, with the option of living away from home; high quality careers advice; an extra-curricular activity and mentoring.

Announcing his first post since he left Parliament in September, Mr Cameron showed his commitment to NCS by saying he would chair a panel of its patrons. He has pledged to make non-military national service a “rite of passage” for every teenager, but does not want it to become compulsory.

Although initial recruitment targets were missed, supporters of NCS insist it is a success. The number of young people taking part rose from 8,500 in 2011 to 58,000 in 2014. The number of places is due to rise to 300,000 by 2020.