You can choose where to get treatment and who treats you; the main issue is whether you get the timeline you'd prefer. Generally, patients will choose a clinical center (to the extent that some hospitals have stopped offering specific treatments, because their stats were never good enough - e.g you need to do a certain number of children's heart ops per year to be good enough to get good stats). If you have the option to go private, either through your insurance or the NHS, then obviously there's another layer of choice.

Most of the private surgeons are also NHS, anyway.