The only hospital which routinely measures Testosterone with every PSA test is UCLH as far as I know, although the oncologist who pushed for this has left, so not sure if they still do that.
It would be useful if everyone who is going on to time-limited hormone therapy on a curative treatment path had a baseline Testosterone test before starting the hormone therapy, but that's extremely rare.
My oncologist and some others add Testosterone to PSA tests when people finish hormone therapy, until Testosterone is back at a stable level. Without knowing what Testosterone is doing during this time, you can't make sense of the PSA level, and it's also useful to know how the recovery from hormone therapy is going.