4 months in advance you are seeing the wrong GP! I could easily switch to a new GP if I tried to make an appointment and they were booked that far in advance.
Can you choose you GP in the UK or are they assigned?
It does depend on whether they have space on their books of course. But by and large, people choose a GP close to where they live, and pick the first one they like the name of on the list at one of the closest surgeries.
The way the system works here is if anything is life threatening, you just go to Accident & Emergency at the nearest hospital.
Otherwise you go to your GP for pretty much everything. You'll have an appointment within a couple of days if it's non-urgent, or told to go in immediately. They perform a basic diagnosis, and either take care of you then and there (minor first aid, basic procedures, prescriptions), or you get referred to a specialist. If you are getting referred, you can say which one you want to be referred to.
And from all of that, you never ever get a bill or see anything relating to accounts. You just get your treatment and care.
Nothing like the states, where, in a similar part of the country (city/rural wise) I needed to make non-emergency appointments 4 months in advance.