So more people from India and Pakistan for example (which many people complain about)? I'm not sure you'd get many Aussie doctors (to use the example above your post) as only someone really desperate to sample British life would want to take a pay cut and work longer hours.
It's the same issue with many other professions. The UK just doesn't pay enough compared to most other english speaking nations, meaning immigrants filling the jobs will be coming from poorer countries, quite possibly with english as a second language.
I have no problem with Indian Doctors, my GP is Indian. But there's more countries than just those 2 in the world, and there are more countries than just the one's in the EU we should be looking at to recruit people.
To me the the Remain group seem to think Leave want to close the door to the world, where as for me it would be opening the door to all those countries outside of the EU and treating everyone the same, whether they're in Europe or South America.