I was wondering how easy would it be to register with a GP under a false name and DOB but with a real address?
I understand this is not illegal since anyone living in the UK for 3 months is entitled to NHS treatment.
It is not illegal to be known by any name you like. A GP can decide for himself whether to register you as a temporary resident whether or not you are strictly entitled to treatment. If you really like your privacy, you'd be better off as a private patient if you can afford it.