It comes as the People’s Vote campaign reveals that more than three-quarters of NHS trusts it quizzed have made no preparations for the UK’s departure from the EU.
They include hospitals in Theresa May’s constituency and in the seats of Dominic Raab, the Brexit Secretary, and Liam Fox, the international trade secretary.
Of the minority of trusts that are preparing, many are worried about the impact on services and about staff shortages, People’s Vote said.
Dr Mike Galsworthy, the founder of NHS Against Brexit, said: “This new polling shows what we all know: that the promise of £350m a week is bust, and all forms of Brexit are costly.
“After all this, we’re left only with a fallen pound, EU staff leaving, extended austerity and preparations for medicine supply shocks.”
And Dr Andrew Dearden, the BMA’s treasurer and Brexit spokesman, said: “Our members have made it clear that they are worried that Brexit could seriously undermine the provision of healthcare in the UK and Europe.
“No one could have imagined the extent of the complications such a result would bring. In light of what we know now, it is imperative that the public has a say in any proposed Brexit deal.”
The Independent has launched its Final Say campaign, to give the British people the crucial decision on any Brexit deal, which is supported by more than 850,000 people who have signed our petition.
The £350m-a-week for the NHS claim was the most controversial made by the official Vote Leave campaign, after polling suggested it was crucial in winning support for Brexit.
Statisticians have repeatedly pointed out it is misleading to claim it is Britain’s EU contribution, because it ignores a rebate and EU funding of projects in this country.
Furthermore, shrinking economic growth because of Brexit will easily wipe out any savings. Some calculations have found that Britain is already more than £350m-a-week worse off.
The YouGov poll – of 322 doctors and 707 nurses – shows a clear shift towards staying in the EU, now preferred three-to-one, up from two-thirds of respondents two years ago.
Backing for a Final Say referendum on the outcome of Brexit negotiations stands at 71 per cent to 29 per cent, once “don’t knows” are removed.
Asked about the promise of a £350m-a-week bonanza for the NHS, 66 per cent of doctors and nurses described it as “deliberate lie,” and 14 per cent untrue but not deliberately so. Just 8 per cent said they still believed it to be true.
The polling comes as the chances of any deal Theresa May secures being rejected – thus increasing pressure for a fresh referendum – remained high.
And leading Tory Eurosceptic Jacob Rees-Mogg insisted around 40 Conservative MPs were steadfast in their opposition to the prime minister’s Brexit strategy – with 39 of them more “tough-minded” than him.