When my husband retired and we were no longer covered by his company health insurance we took out a policy with National Friendly (I think they may have changed their name now). This policy was part health cover, part savings policy. If you made a claim, part came from insurance cover and part from savings but if you didn't, the savings built up. Not sure if they would offer the same deal now.
We both used it a few times over the years for minor things but still built up quite a nice cash sum. We cashed it in recently for the following reasons:
Major health problems are dealt with far better on the NHS. I was unable to use my private insurance anyway when I had cancer, as the NHS hospital couldn't (quite rightly) split my treatment for part private care as it would've been an administrative nightmare. My insurance company wouldn't cough up for reconstructive work that I had to have (one of my operations was part cancer removal, part reconstruction).
I had a string of awful private consultations with clapped out semi retired behind the times consultants. One where my results were sitting in a cubby hole for more than a week. No one thought to tell me they were back.
We could afford now to pay for anything minor if there was a lengthy waiting list and most minor things do not involve lengthy stays in hospital anyway, thereby negating the need for a comfy private room.
All of the times I have been into a private hospital, I've been too ill or not allowed to partake of the wonderful menu/wine list!! And too groggy to know where I was anyway.
So weighing all of that up, we decided it wasn't worth it for the cover provided.
They won't cover existing health problems anyway, so I would put the money you would've forked out for insurance into a savings account for anything minor. Anything major is dealt with more expertly on the NHS in my opinion.
Slightly off subject, we've done the same for one of our dogs. The insurance premiums (reasonable to being with) were ridiculously high for him because of his age (when you may need it most! Not stupid insurance companies are they.). So we cancelled it and have put the amount we were paying into a savings account letting it build up to offset anything he may need. The annual premiums were so high they were going to cost almost as much as his cover! Crazy.