The NHS pay cap is unfair, unpopular and dangerous to patient safety, bodies representing 1.3 million health service staff have warned Theresa May, urging her to ditch the policy in the Queen’s speech.

The plea from doctors, nurses, dentists and other health professionals comes as the prime minister faces intense pressure to scrap the cap, introduced in 2010, which has limited NHS staff to 1% pay rises or below. It is legislated to continue until 2020.

In an unprecedented joint letter, a range of health trade unions told May the policy was among the reasons why the Conservative party lost its Commons majority in the general election earlier this month. “By your own admission, austerity, and a lack of investment in the public sector was a significant factor in the general election result,” the letter says. “Many have said that the pay freeze in the public sector was in part to blame for your failure to secure a parliamentary majority.”

The letter follows the admission last week by Stephen Crabb, the former work and pensions secretary, that he lost votes because of Tory spending cuts. After years of wage restraint, teachers, nurses and local council staff all deserved a pay rise, he told the BBC.

NHS staff have become increasingly angry about the pay cap as inflation has soared from 0.3% in May 2016 to 2.9% last month, its highest level in four years.

The unions are urging May to “mark a clear change in direction” with regard to the public sector in the Queen’s speech on Wednesday, when the monarch will set out the government’s legislative programme for the new parliament.

“The public sector pay cap has forced professionals out of jobs they love,” the letter says. “Those who stay are overstretched and under pressure to do ever more with less.”

NHS Providers, which represents hospital trusts, recently revealed that staff were quitting to stack shelves in supermarkets.

“The longstanding cap stands in the way of recruiting and retaining the best in healthcare,” say the signatories, which include the Royal College of Nursing, the British Medical Association, and the British Dental Association.

Unison, Unite and GMB, which represent nurses, paramedics and other NHS staff, have signed the letter, alongside Managers in Partnership, the union for about 7,000 senior managers across the service.

“[The cap] is having a profound and detrimental effect on standards of care for people at a time when the NHS is short of staff across every discipline. This is alongside an uncertain future for EU nationals working in health and care”.

With the NHS about to turn 69 on 5 July, May should mark its 70th year by scrapping the policy of real-terms cuts in income as a way of showing how much she values NHS staff and patients, they add.

Jonathan Ashworth, the shadow health secretary, said: “Our hardworking NHS staff deserve better than to be taken for granted by a Tory government content with demanding more for less.

“Theresa May’s utter disregard for health staff was clear when she shrugged off the idea of nurses using food banks. However, the election result proves that the public will no longer tolerate this government’s neglect and disregard towards those who care for us at our time of need.”

Each 1% increase in staff pay would cost the NHS an estimated £500m.

In a brief statement, the Department of Health (DH) said: “As the secretary of state has made clear, the support and welfare of NHS staff is a top priority as they do a fantastic job. The government is committed to ensuring they can continue to deliver world-class patient care.”

DH sources stress that this year’s 1% rise was based on a recommendation of the NHS pay review body. However, it also warned that 1% rises were no longer sustainable and that its remit had prevented it from suggesting a bigger increase.

Meanwhile, the BMA is urging ministers to prioritise the needs of the NHS in their Brexit negotiations with the European Union, which begin on Monday.

“Leaving the EU poses several risks to healthcare across the UK, not least in its staffing as almost half of the 10,000 doctors working here are considering leaving in light of the referendum result, said Dr Mark Porter, the union’s chair of council. “These doctors have enhanced the UK’s medical research, brought expertise to the NHS and higher education, and filled shortages in specialties which may otherwise have been unable to cope.

“While we welcome the government’s pledge to provide certainty for EU nationals working in the NHS, the time has come for it to deliver fully on those repeated promises by providing them with permanent residence in the UK.”