Ministers plan to overhaul the system in September 2017 and charge those studying to be front-line health workers for their degrees.
But in a letter to the prime minister, more than 20 groups led by the Royal College of Nursing said it could put people off working for the NHS.
The government said the changes would create 10,000 more training places.
The move comes as nurses and doctors gather for their annual conferences.
The RCN conference gets under way in Glasgow on Sunday, while the British Medical Association meets in Belfast from Monday.
Currently student nurses, midwives and other staff such as physiotherapists are entitled to bursaries of £4,500 to £5,500 – on top of a grant of £1,000 each year during their training. The course fees are also covered.
Those studying for advanced roles, such as district nurses, could also be affected.
But the government has proposed scrapping these and introducing university fees to bring health staff in line with other students.
Ministers argue that the move will lead to an increase in nurse students – of about 10,000 – as applicants for courses currently outnumber the places available by two to one. This is because there is a cap on places.
But the letter, which has been signed by the BMA and the Patients Association, warned it was an “untested gamble” and urged the government to halt the plans to properly consider their impact.
RCN general secretary Janet Davies said that as nurses spent about half of their degree course working in the NHS it amounted, in effect, to “asking people to pay to go to work”.
“It feels very wrong,” she added.
She also expressed concern about the state of the NHS.
Figures released last month showed NHS trusts in England recorded their worst ever deficit in 2015-16 with an overspend of £2.45bn.
Meanwhile, performance targets for A&E, ambulance services and cancer care are all being missed in England. Other parts of the UK are also struggling.
An online poll of UK nurses by the RCN found only one in 10 thought the health service was able to meet demand.
The BMA has released a survey of its own – of 1,240 members of the public in England.
It suggested eight in 10 were worried about the future of the health service.
The poll – carried out by Britain Thinks – also found fewer than one in five trusted the government with the management of the NHS and only one in eight believed it was getting the money it needed.
More than half felt the NHS was going to get worse – with just 14% saying they would expect it to get better.
BMA leader Dr Mark Porter said a long-term plan was needed to address the “crisis” in the NHS.
“People are increasingly concerned about the future of a health service that they know is under unsustainable pressure.”
A Department of Health spokeswoman insisted the plans for students would lead to more people studying as the caps would be lifted.
But she urged those concerned to take part in the continuing consultations on the issue.
A Department of Health spokesman said: “We need more home-grown nurses in the NHS because they do an amazing job caring for patients, but currently two-thirds of people who apply to become a nurse aren’t accepted for training.
“Our plans mean up to 10,000 more training places by the end of this parliament, with student nurses getting around 25% more financial support whilst they study.”
On the wider concerns about the NHS, the Department of Health said the extra funding being provided this Parliament would “transform services”.