Here are my timelines. I'm in Ireland which is a different medical system to the UK. I have supplementary private cover which is common here but public timelines and treatment is not much different for cancer.
May-Aug: Took 12 weeks post op to confirm persistent PSA. Two PSA tests at 8 and 10 weeks plus 2 weeks to get PSMA scan and meet consultant urologist who did my operation.
September: Referred to two medical oncologists one for radiotherapy and the other for hormone therapy. Met both within 2 weeks. I would have been put straight on HT but had to had a biopsy of a suspect lymph node which was done in 2 weeks.
October: Biopsy was negative so first HT jab was beginning of October.
November: Started chemo at end of Nov. Radiotherapy will follow this.
So here it takes about 2 weeks to see a consultant and about the same (or maybe a month) to get a PSMA scan. My advice is to call around to get to see a medical oncologist and a PSMA scan as soon as possible. You would be amazed what making calls and explaining your situation does. I have always found that people in the medical profession understand when you are a genuine case and will try to accommodate you asap. Don't take what the computer says as the answer until you talk to people. Just this week I got a chemo session and a blood test pulled forward 2 weeks from making a few polite calls and talking to people.
The PSMA scan really is required to give any chance of a diagnosis of your case. Normally you want to do the PSMA scan before HT so that any suspect areas can been seen before the HT shrinks them. Before my operation I paid $2k for a private PSMA scan which was not covered by insurance is maybe that is an option? The oncologist will be able to review your scan so you can do this separately if it will be quicker. The urologist who did your operation should be able to give a referral. Apart from waiting for a scan you should be on HT asap and then other treatments (normally radiotherapy if no distant mets identified) will be on top of that.
If you do not have any treatment your PSA will continue to increase so I would not be at all happy waiting over two months in your situation. To put in context, those 2 months I met 2 consultants had a PSMA scan, a hilar node biopsy, started HT and had a plan for chemo and radio. Since going on HT my PSA has gone from 0.8 to undetectable and my other treatments are front loaded to try to cure it with obviously no guarantees.
Best wishes and good luck getting things organized. It is a stressful time but it gets a bit better once you get a plan in place.