I had half of my face stop working and had recently moved (and not yet registered with a new GP.)
I went to my previous GP and they turned me away (no longer in the catchment area) so I eventually ended up in A&E.
It doesn't help that the local ones near me at the time seem to live in the past... No call queuing or online scheduling, have to call within a 1 hour slot for appointments (so that's an hour of busy tones, oh well let's try again tomorrow)
A GP practice is a private business, usually all the GPs are partners just as lawyers or accountants would be, and the staff such as receptionists are not NHS employees, they work for the practice. If they can't bill the NHS for their services because you're not on their books, they aren't interested. It is surprising perhaps, but many people in the UK are unaware of this, they think the GPs surgery is directly a branch of the main NHS, and get angry about "privatisation" without realising that GPs have always been private.
There is a hospital in my town but it's A&E was closed in 2007 IIRC, so if you can't see a GP you can't even go there, you have to drive a couple of towns over.
I'm not sure they are. Here's the (admittedly complex) book: https://www.england.nhs.uk/publication/primary-medical-care-...
Note this is only for England. I have no idea about the rest of the UK.
If you're outside the catchment area the GP can chose to offer full service; offer limited service; or offer no service.
The full service includes things like home visits, so it's not surprising that GPs seek to limit that to patients who live far away.
For people who are travelling there are temporary registrations, but also GP walk in centres and GP out-of-hours services.