Yes, you are very lucky compared to some people, Quite often busy surgeries are booked a week or even 2 weeks ahead, and even if you're about to die you have to wait for the "emergency" slot at the end of surgery time.
I think I'm even luckier - I just turn up before 10:30 and wait to be seen. The drawbacks with this system are that sometimes you have to wait a very long time (though I've waited just as long in some places with an appointment), and that on occasions the GP is very busy. One Monday my GP saw 56 people at morning surgery! I doubt if he was giving as good a service at the end as he was at the beginning. (I went away and came back 2 days later).
Also, you can always see the GP you want to see - just turn up on the mornings he is in. The practice runs 2 surgeries, and if pushed you can see the doctors at either location, but the one furthest away doesn't have my paper notes. Afternoon/evening surgeries are appointment only, and reserved for people who work and school children.
You have to make an appointment to see the nurse unless you're an emergency (bad wounds, or suspected concussion etc). You can almost always get one within a week.
The surgeries also have visiting physiotherapists, podiatrists, dentists and opticians. Most of those are at the main surgery, not my local one, but it's only 9 miles away. I think I could possibly do it by public transport, but it would mean a lot of time to get there and get back - about 5 hours. As long as I still have a car I'm okay.
Both surgeries also dispense medications.
Viv 8)