Twice I went to my GP in London and was referred to the blood test centre at the hospital.
It was a walk-in, no appointment needed (just queued for 10 minutes), and the sample took less than 5 minutes. All I had to bring was my referral sheet from my GP.
I could have gone any time during their opening hours (M-F business hours, I think) but both times I went the next day.
They sent the results back to my GP within 5-7 days.
The only thing that bothered me about the whole process was that they'd only call if there was anything abnormal (there wasn't so I received no call) but I was able get a print out of the results for my records.
Even though my GP didn't proactively offer me the referral, I wanted to be tested since there are thyroid issues in my family. She happily wrote me the referral without trying to convince me otherwise (although she did assure me it was probably nothing to worry about). The tests came back normal, but I was happy to be tested and know for sure it's nothing to worry about.