Hi Farmgirl.
We are in Scotland, so the NHS rules may not be the same here, but...
I had some trouble registering as I was not employed or registered as looking for work (I'm in the "self-sufficient" category), but now am happily in a practice I really like. You will register with someone in your local area - NHS Inform's website has an option to search by post codes. They may or may not accept you as new patients (they are not required to) if they are a full practice, but when I spoke with NHS Inform about our particular problem I was given the names/contact info for three surgeries within a mile of our home, and told that if none of them would accept us to call a dedicated number they gave us and we would be assigned a GP who would have to accept us. Thankfully, we didn't have to go that route.
Our surgery wanted something with our address on it - utility bill, council tax notice (that came pretty quickly), or a lease. You'll need to bring all your passports, of course. In our case, we were asked to go pick up registration info, and were given an appointment at that pickup to come back to meet with the intake nurse about a week later. Spoke with her for about 20 minutes each, and we're registered. It's been less than two weeks and I'm already getting mail from the NHS asking me to have screening tests done (due to my age).
Health insurance is interesting here. Most of the policies I found when looking for one for my daughter have you use the NHS for emergency care, and go through the NHS surgery GP for the first contact. We settled on Vitality Health, which has a tele-medicine component - you can do a video consult with one of their GPs for minor things and have them phone in prescriptions at no charge. For most of the other insurances we looked at, if you need specialist consultations, tests or MRI/CAT scans, or non-emergency hospitalization, that's where the private insurance kicks in. At least with the plans we found to look at, there is a "core" cover for hospitalizations, and then you add additional services on a sort of cafeteria plan (they are really big on selling cancer care, which is a bit odd to us and makes us wonder about NHS cancer care?).
We took out a plan for my adult daughter that covers everything the company offered (other than dental) and it's costing us just a bit over 50 pounds a month. Beware the pre-existing conditions thing - unlike under the ACA, if you've been treated for/diagnosed with/had symptoms of anything in the last 5 years that is one of those ongoing things (like diabetes or high blood pressure) you will probably not be covered for it on the policy.
We are sending in for my daughter's residency card, so we'll find out in the next several months if they consider Vitality Health Insurance (which pretty much mirrors most of the rest of them) to be "CSI" or if we'll have to go back to one of the really expensive "expat" insurance companies. Could never find any serious guidance other than that whatever insurance you choose, it has to cover you for the "majority" of any medical needs you may have.
Hopefully someone down in England will chime in and give you more localized advice. Good luck with it. (We really like our surgery, very efficient, very pleasant place. Not at all like the doctor's office factory we were used to "back home".)