That actually depends on implementation: there are countries where private practices practically don't exists besides people in the 1% who pay for the convenience and attention; and countries like Poland where a lot of people have work-provided healthcare benefits which mostly decrease wait time for specialists - very limited subset of healthcare overall, there are no common private cancer treatments etc.

I don't think there is country where people buy very broad private coverage after having mandatory public one, so the scenario you're trying to present wouldn't happen: there would not be enough jobs for "all the best doctors" in private sector.