Certainly not all, as my wife doesn't (she tried it, and among other reasons, decided she would rather have a life), nor do some others I know of. It's not an easy option as contrary to common belief consultants are normally working fulltime for NHS and they can't bunk off early to go and do their private work. Hence much private work is done after hours or at weekends.
They'd get into a lot of hassle at the (NHS) hospital if they did (ignoring quite how they would delay the operation, they don't control the list directly). 'Breaching' (as in breaching the 18 weeks time limit) is a big deal in the NHS, as it's one of the key criteria hospital are judged on. So it's also a key criteria on which departments and individual consultants are judged. So if were breaching, they would need to justify why, and show waht action they were doing to avoid it in the future.
The normal way of dealing with a likleyhood of breaching is to run extra operating lists (which will quite often cost more, but not as much as paying for it privately, as they may well have to pay for consutlants and other operating staff to work extra sessions)
The private hospital doesn't care, if the NHS has decided to use their services.