Re. hospitals, it depends. King Edward VII a few streets away from the London Clinic is one of the hospitals the Royals use, but DoE had heart surgery at Bart's, and PoW, not to mention others have had children at St Mary's.
My mother was at the London Clinic for various treatments, and King Edward VII f(or that matter). Many private consultants with many different specialties work at both. Usually it's a case of going where a specific doctor operates.