I have some knowledge of rothbury surgery having done a registrar job there 20 years ago. The surgery was in converted housing over 2 stories and totally inadequate for modern practice. It was never owned by the NHS or signed over to the doctors, it had always been owned by the partnership. They had started the process of designing the new hospital when I was there and it looks like it took a good few years after that before finished. It was the doctors who started that recognising their current building was not up to the job. When they moved into up to date building then they pay rent and service charges to be there. The old building was always theirs to do with what they wanted so if they sold prime plot right in centre of a pretty town I’m sure it will have made them money, but it was their asset to with what they wanted. The branch surgery in longframlington also had never been owned by the NHS but was built by the partners again.
The info I got from a certain person's estates manager (you probably know who) was that it originally was an NHS asset that was essentially gifted to the GP practice.
As you say it is in a prime town centre location and was sold for a very tidy sum. Or rather several as they split it up.
Yes, they do pay rent to be at the new(ish) hospital but that's basically just passing money in a circle as that money is largely from the NHS!