I can't answer for the BUPA prostate pathway as no knowledge or experience.
When I had a high PSA, my GP had already referred me to the local hospital before he called and told me. However, I had BUPA cover, so I asked to be referred privately. He was quite firm - twisted knee, strained shoulder - fine to go private. Cancer diagnosis - you want to be on the NHS. I didn't really understand that at the time, but he did do the private referral too. The NHS referral appointment came through first - mpMRI scan, followed next day by an all-day appointment with the urology one-stop shop (which was newly opened then in 2018), so I cancelled the private referral.
Now that I do a lot of support, I completely see where my GP was coming from. The NHS has mandated MDT (Multi-disciplinary team) working for ~20 years. This was specifically to counter the issue of surgeons who run the diagnosis pathway from being over-optimistic about their own treatments, and to ensure patients were offered all viable treatments. It's not perfect by a long way, but many patients do avoid prostatectomies now which were never likely to be successful. Indeed in England, it's around 2/3rds go for radiotherapy and 1/3rd surgery now (Wales is 3/4 and 1/4 respectively).
If you go privately, there's no requirement you will be managed under an MDT, and some patients are only ever under the urologist they were referred to for diagnosis, and there may be no consideration of other treatments. I have seen a number of such patients who did this and went the surgery route, where it wouldn't even have been considered on the NHS because too likely to fail and other options were better.
This will depend massively on where you are in the UK. My hospital now runs a one-stop-shop for urology cancers with the MRI done the day before. Some even do the MRI on the same day. Of course, there are still hospitals doing every test one-at-a-time with 3 or more week wait in between. You can ask your GP to refer you somewhere else in such cases. Some places run the RAPID diagnostic pathway.
If you do go private, I would try to ensure you are going to be under an MDT.
If you go NHS and have private cover, you can tell them that you would be interested in private procedures and then they can advise if any particular private diagnostic or treatment option might be relevant, but still stay under the NHS MDT. Clinicians are not permitted to mention private procedures during an NHS appointment unless you've explicitly asked them to do so.