I guess the nationwide (near) monopoly on healthcare, the National Health Service (NHS) will have to start paying market prices for nurses as well as start more training programs instead of relying on importing cheaper labor.
To say that there aren't enough Brits to do the nursing in a population of 60 million is ridiculous.
The elites in Britain (and perhaps the entire wealthier countries of the EU) basically didn't seem to care (and perhaps encouraged) the importation of labor to drive down market prices of labor both for untrained as well as trained labor. In the case of healthcare, it is simply a means for the government to pay less for healthcare than they would if there were a competitive market instead of their monopoly.
It wasn't absolutely necessary to have the free flow of labor as part of the EU trade agreement. For example, NAFTA between US, Canada, and Mexico does not allow for the free flow of labor, only the free flow of good. I can't think of any trade agreements that the US has with other countries that allows for the free flow of labor. The free flow of labor benefits elites (think professional hiring nannies, government running health services, companies running factories, firms running high tech needing software developers) to help depress wages. Now the elites of Britain will have to start paying market rates for their labor.