As Drakey has said, there is an axe-grinding issue here so not sure whether you want my advice or not.
The problem with having a civvy GP as well as those in the RN is that you will have 2 sets of notes, be referred to separate hospitals, etc - and your civvy GP cannot provide the occupational health that you will require. If the civvy picks something up then you will have to bring it to the attention of Naval Medical Services and there will be issues at that stage.
Everything you say to a Naval MO is in confidence - the only bits of your records which are 'Staff' instead of 'Medical' are your vaccination status and medical category. Naval MOs, like all UK doctors registered with the GMC, cannot divulge further without your consent. The only exception to this is at sea - the Captain has to know everything and has a right to know. That is a bit strange, but I'm sure you can see the reason for it.
You won't believe me, but the medical care provided to those in the forces is much better to that in the NHS - several reports confirm this. Just because you have had a run in with a doc doesn't mean we're all the same!
Incidentally, base ports and most establishments have CMPs - Civilian Medical Practitioners - civvy GPs employed by Pusser to see patients.
In summary, you are not entitled to routine NHS primary care (emergencies are ok). If you develop some condition or other then you will have to let us know anyway (to do otherwise would be, frankly, a bit stupid and also against Naval (now Military) law).