Ross,
It depends who is supposed to be doing your PSA tests. At this point, it will be the responsibility of your oncologist, but they may have asked your GP to do it every 3 or 6 months. Do you get a copy of the letters from your oncologist to your GP?
Sometimes my oncologist gives me blood test forms, and sometimes I ask my GP. The receptionist can't order blood tests, so no point in asking them (unless the protocol at your surgery is they go and ask the doctor). I can message my GP directly through the patient portal, and he prints off a form and leaves it to pickup from reception (I've suggested emailing it, but that's a bridge too far so far). I always justify why I'm asking, and he's never refused. Usually, it's in preparation for a consultant appointment, but I have asked for a couple of testosterone tests (one when I saw a reversal of some hormone therapy side effects and wondered if an injection hadn't worked, and the other because my consultant has asked for them now that I'm coming off HT). I have asked for liver function tests a couple of times to make sure my liver is coping with Tamoxifen, because it didn't to start with. My GP has always done them, but that's probably because I had a valid justification. If you just ask for a test without having a valid justification, I can imagine they might say no.
The last consultant letter says he wants 3 monthly PSA and testosterone tests, and it seems to be up to me to ask the GP when one of these is due, but he's given me the test form for a test just before my next consultation 10 months after the last one, and that one has a load of other stuff on it too which I've never had tested before (like bone profile) as well as blood counts, liver, kidney, PSA, testosterone, etc.
So you could ask the receptionist to book you a doctor appointment (assuming you can't do it on a patient portal), and when they ask why, say it's to ask the doctor for one of your routine PSA tests.
I think NICE says you should be tested at least ever 6 months - my consultant has asked for it every 3 months (although in practice, I have slipped that to 4 or 5 months to it lines up with consultations).