If we're talking about robotic surgery, I'm afraid that even that probably won't be true! I had my nephrectomy operation (kidney cancer) privately - work medical insurance - and the private hospital (like most) did not have a Da Vinci robot. Had my surgery in the local big NHS hospital (which did had a robot) where it was performed on a Saturday by the consultant urologist I was seeing privately, who was also the head of urology at the hospital. Like many NHS hospitals, it makes extra money by renting out the use of its operating theatres at the weekend to private patients. It doesn't have private rooms, so I was on an NHS urology ward, eating NHS food (which was absolutely fine). The only thing I got which an NHS patient probably wouldn't have was a visit from the consultant on the Sunday to see how I was doing. Other than that, my surgical treatment was exactly the same as an NHS patient.
Cheers,
Chris