Ive been here almost three years and only just started learning the NHS system :/
Registration at the GP should be painless - I took my passport, visa proof, and two proofs of address (they really wanted bank statement and something like power bill or council tax) and filled out a questionnaire and in 15 minutes I was sorted. I think I still had an old number from when I last lived here in 1999, though.
In the last few weeks I have had to go to the doctor due to a back problem. I have called twice for an emergency appt and gotten in same day both times, although for really a 5 minute chat with stressed doctor. I also made and kept another appt. He put through a referral for an MRI for me first thing, which was arranged separately the next day (but for a date 3 weeks later). I missed the appointment as my back got worse and I couldnt actually lie flat, but rescheduling was easy, although another two week wait.
To me it seems like it works like a well-oiled machine, but expect the amount of doctor time to be reduced. The GP is almost like a triage person to sort folks out into the different paths and specialties. Also, I didnt realize that they can do phone consults here, so I could have arranged one of those instead for some stronger/different pain meds rather than going in and then the office can call in the prescription.
It seems really intimidating and overwhelming if you are used to the US system, but so far it has worked ok for me. I will admit that I was apprehensive too about signing up, but thankful I did when I did!