In my experience some GP surgeries perform "physicals" when you register as a patient, but I have not found this to be the norm in the recent past. This may be due to funding constraints, or may also be due to the belief that, in fact, most physicals ARE a waste of resources for the under 50's. I have experienced both systems and would say that my annual physical in the US was a waste of resources for someone in good health, in that it just confirmed that I was in good health.
In the case of who operates on you in the UK, well you probably won't have a choice, nor will you have a choice of hospital. It is possible to research consultants who practice both privately and on the NHS, pay for a private appointment and see if it is possible to be seen by that consultant at their NHS clinic. This used to be know as queue jumping.
Perhaps you could take out some form of private insurance that would allow both an annual physical and access to consultants, but rely on the NHS for regular GP appointments?