In my experience, registering with the GP was quite easy and straightforward. I got an application form from my local GP (personal details, contact information, past medical history) and returned it with proof of address and a copy of my passport details page and visa page. A week later, I got my new NHS card in the mail.
My GP was hesitant to refer me to a specialist at first, and I needed a few trips to see her with continuing symptoms before I got a referral. My medical records were finally received at that point from my previous GP, and whether or not those had any bearing on getting a referral, I'm not sure. The referral was requested immediately; however, since my symptoms were bothersome but not considered urgent, I had to wait several weeks before I could see my specialist.
(to be fair, when I was asked my choice of specialists, I opted for one at the top hospital known for my condition...I could have seen a specialist at my local hospital and that may or may not have been quicker, but I can't say for sure)
About prescriptions though, I got those immediately. The GP asked what I was on (name and dosage) and wrote out replacements right then and there.