A non critical medical appointment during quarantine would really not be OK, if you are skipping care now it can certainly wait two weeks. Going to a medical facility for a non emergency appointment puts *everyone* at risk. I'm sorry but things like that make me angry, so many people have been waiting for surgeries and essential care and you think that it's fine for you to just hop the line and go private while you may be contagious if you pick up covid on the plane. A doctor should refuse to see you during quarantine for any routine/non emergency care. (In A&E they have the covid areas separate to everyone else so they are prepared for it.)
All of which assumes that private medical care offices are seeing anyone at present. Our NHS GP is still doing telephone triage only. The pharmacies are open, and have been during the epidemic emergency. To get a NHS prescription here we have to physically walk to the GP to pick up the piece of paper and hand-carry it back to the pharmacy, and then come back a couple of days later to pick it up. (All the prescriptions are filled at a central service center, I believe.) Thankfully, we have not needed to get a medication or refill medication during the pandemic - I have no idea how they are handling that.
Normally I ask for and have had no problem getting Naproxen. Not strong enough at all times, but better than the paracetamol (sp?) that seems to be recommended for everything. Paracetamol (tylenol - acetaminophen) is bad for people with liver issues, so if you have those you'll want to be sure to let your doctor know. It is also ineffective for low back pain and similar:
Paracetamol is ineffective in the treatment of low back pain and provides minimal short term benefit for people with osteoarthritis. These results support the reconsideration of recommendations to use paracetamol for patients with low back pain and osteoarthritis of the hip or knee in clinical practice guidelines. https://www.bmj.com/content/350/bmj.h1225 I can vouch that it's like taking a sugar pill - does absolutely nothing for spinal nerve pain. At all. The Naproxen does help some, though.
When looking at private insurance, you might investigate those that have a telemedicine option - you have your consultation with your doctor via the internet or phone. I know Vitality (insurance) has that option. We got that for my daughter when we came over here because it was relatively cheap, met all her regulatory needs, and had that service. We haven't used it because we never got a smart phone - and it has to be one with a specific operating system to work - and actually never really needed to use it. Most of the UK insurances I was looking at here require that you go to your GP first and they then pick up costs if you want to go private after that point. My retirement package from the US includes insurance that has a telemedicine option, but because the doctors are in the USA it's no good for any prescription medications over here. One has to have an in-country doctor to prescribe medication (as is the case in the USA). My USA plan also allows me to see any private doctor and get a reimbursed rate, or to use one on their list (I think that's all there actually is on their list in Glasgow - one!) for a substantially higher reimbursement.
It might behoove you to do some research on your private insurance options, as there may well be one that solves your problem. Good luck!