I have needed medical care a few time in Britain, mostly for skin/pressure issues.
I have travel insurance of course, but I haven’t used it as in Britain (and Italy) as i have been able to access the local health system usually. Italy, especially Florence, is pretty good once you figure out how to find help. Many doctors do home visits if you’re in the city centre, which is great. In my experience they were all women. Britain is much more complicated. If you go to a hospital out of London you might get good treatment, and they will probably give you medical supplies for free (at least they did for me as an Aussie). They gave me “temporary resident” status. Seeing a doctor in London can be tricky. Normal doctors are generally tied to particular areas and residents only. Private doctors are available but the quality is incredibly patchy. There are also “walk-in” clinics but these can be unhelpful as well with long waits. Sometimes it’s hard to see, just an overworked nurse. My worst experience was with a private clinic in a good suburb, overpriced and incompetent (Roodlane Medical, 58 New Broad Street London). My best experience was seeing a doctor privately in a poor inner suburb which was cheap and helpful. Best of luck!
I also visited a clinic in Holborn which is convenient if you’re in the inner city. They were adequate but had a lot of weird industrial health and safety rules from an Aussie’s point of view. Access is at the front left via a slightly dodgy sliding door Holborn Medical Centre 64-66 Lambs Conduit Street, London, WC1N 3NA, Tel: 020 3077.