Various reasons, mostly you're guaranteed a private room and in some cases, you get to jump the queue - not if it's an emergency or life-threatening, of course, but for some elective surgeries and heavily used things like hip replacements where there's a wait-time. You're mainly still using NHS hospitals and personnel and (I've heard) it can feel bad due to a general dislike of queue-jumpers by doctors and nurses. It's very much a tiny (wealthy!) minority . who do it. For the most part there are only short (days) wait times under the NHS and none at all for anything life-threatening.

Hard to say whether I'd pay a co-pay or how I'd feel about it - It's too alien to me, the concept of doing anything other than just turning up at the GP or hospital and expecting to be seen. Paying for that is... I can't compute, tbh, it's not part of my regular mental processing