Important to note the "The NHS" is not a single entity and the GP practice is likely a private entity owned in partnership by the doctors. There are a number of reasons why individual practices can refuse to register.
I went to see my GP and the lady at the reception told me they no longer book visits at the reception and I had to use the app. Here's the privacy policy https://support.patientaccess.com/privacy-policy They reserve the right to pass your data to third party contractors and to use it for marketing purposes. There is the obligatory clause on regarding the right to be forgotten, but the AI companies claim it is impossible to implement.
Take your point about LLMs though.