Around 60,000 NHS staff - 6% of the total workforce - are EU nationals, and 4.1% of England's GPs in permanent roles were trained in the European Economic Area (EEA).
BMA leaders warned that 'recruitment and retention of doctors' was a key concern. The UK is expected to leave the EU at 11pm on 31 January - around three and a half years after the 2016 referendum saw voters back Brexit by 52% to 48%.
Prime minister Boris Johnson is expected to make a statement as the UK quits the EU, to say that the move marks 'not an end but a beginning'. Although the formal departure from the EU takes place today, a 'transition period' will mean that the bulk of EU rules remain in place until December.
The BMA - along with other medical organisations - has repeatedly voiced concerns about the impact of Brexit, in particular warning that a no-deal departure would spell disaster for the NHS.
In a Brexit-day message to EU colleagues, BMA chair Dr Chaand Nagpaul said a no-deal situation remained a 'very real risk'.
He added: 'Freedom of movement has allowed talented doctors and other healthcare staff to travel, work and teach across Europe, letting those from the EU to both contribute to and learn from the NHS, while UK-trained clinicians have been able to share their skills in other European nations.
'Our message to our European colleagues – and indeed those around the world – is a clear one: you are welcome here. While the UK may be leaving the EU, Britain’s doctors will remain very much part of Europe’s tight-knit medical community. We have far more in common with our European colleagues than is different. We are one profession bound by our duty and desire to provide the best care to our patients across the continent.'
BMA Northern Ireland chair Dr Tom Black warned of 'issues still to be resolved' - calling for guarantees that cross-border healthcare arrangements on the island of Ireland would continue, and raising questions over how EU-funded services could be maintained.
He said: 'Today marks the beginning of leaving the EU and while this cannot be changed, we remain the only part of the UK to share a land border with Europe, and that unique position presents ongoing uncertainty and insecurity for the delivery of health here.
'One of the key issues in resolving our health crisis will be the recruitment and retention of doctors and other healthcare staff. With just 11 months to finish negotiations on our future relationship with the EU, there are still issues to be resolved: current cross border health services must be able to continue and indeed expand; the practicalities around the transport of medicines over the border still needs to be resolved and replacement funding for EU funded services will need to be found.'
Dr Nagpaul warned it was vital to ensure 'the supply of vital medicines is not disrupted, protecting collaborative relationships with our neighbours on medical research, and introducing a flexible immigration system that will allow health and social care staff to respond to needs across Europe'.
The government said last year that even in the event of a no-deal Brexit there was 'no question of EU staff needing to reapply for their own jobs'.
A letter from health and social care secretary Matt Hancock to health and social care staff last year said that his view, along with that of the home secretary, was that: 'If you are an EU citizen working in the NHS or social care, we really want you to stay.'
GMC chief executive Charlie Massey said: 'European doctors make a vital contribution to our health service and systems are in place to ensure they can continue to do so in the years to come.
‘The registration status of any doctor already holding provisional or full registration in the UK won’t change due to Brexit. Any new applicants from the EEA will be able to join the register in the same way throughout the transition period. It is crucial that doctors are not deterred from practising here and as a regulator we will do all we can to support the retention and flow of doctors into the UK.’