It seems a mess at the moment; not so many years ago, I could wander down to the surgery between 8.30 and 10.30 am and wait to see a doctor (which never took more than 30 minutes or so) or book an actual appointment if it wasn't urgent. Then that was deemed 'too popular' (in my limited experience, it was busy but not excessively so, but they would turn people away if there were too many people to see).
Personally, I don't think our obsession with a 'free' NHS is helpful, but everything seems very conservative. We seem unable to approach primary care differently (not to mention the wholesale change to preventative care, but that involves cultural changes). The medical establishment is very orthodox (if you want to pickle their onions, ask a group of senior medics involved in training why a successful nurse can't become a doctor, seriously, they turn puce and start to splutter like a 2CV on a cold morning). Since I was involved in research around this, not much has changed, mostly because the establishment won't change the way (and who) they want to train, or the processes they follow. They'll always agree that being a junior doctor is terrible but immediately pivot to how they 'benefitted' from the character-building (and often patient-killing) experience. There's a weird thing about 'resilience' they mutter on about. None of this is a uniquely British phenomenon, of course.
There doesn't seem to be any real planning for how we maintain any kind of service in the future given the rapidly increasing health burdens and costs; mostly because we have a government only interested in selling bits off to their friends, of course.