It used to be able to draw on the EU's liberal, if tight, supply of healthcare professionals. But not any more.
Most of the non-British healthcare workers in the UK actually came from outside the EU. The EU has huge problems recruiting medical staff as well. Getting an optician's appointment in France can take over a year, for example. The French government is even considering taking powers to compel new medical graduates to practice in certain regions (the "medical deserts"). Brexit has made little difference there.