Private hospitals don't have acute facilities such as A&E - you would be transferred to the NHS if that was necessary. (This applies to some NHS hospitals too.) Most NHS hospitals have a private ward and do private procedures, and that might be particularly relevant if the procedure had a higher that normal likelihood of developing an emergency situation during the procedure, as you could be treated on-site with acute facilities.
Another problem for private patients is that no followup will be arranged for things like ED, counseling, continence, access to CNS, etc. In theory, you can transfer back to the NHS for this, but it's sometimes difficult to get back onto the NHS systems if your main treatment wasn't on the NHS. It may also be difficult to get medications afterwards on the NHS. This should all work as patients have a right to have parts of their treatment on the NHS and parts privately, but it's not as easy as it should be.
As others said, unless you have a high Gleason or high risk in some other way, a delay is rarely significant, anf hormone therapy can be used as a holding strategy if you wish.