You are aware that people who pay for private healthcare also pay for the NHS through general taxation and national insurance? Given that they likely have a higher income to afford private, they are also probably contributing more towards the NHS than the average person. Not only that, but by removing themselves from the waiting list for the first (hypothetically botched) operation, they would have presumably initially saved NHS time and resources and shortened the waiting list for others. Since private healthcare is designed to help people, not harm them, it would only be bad luck that they wound up needing further treatment after they received private treatment.
So, you think these people shouldn't be allowed to use the NHS, despite paying more for it and being less likely to use it? I would be careful what you wish for. If people with private healthcare should be barred from NHS care, they should have no obligation to pay for the NHS, either. Why should they be forced to fund a service that they are not allowed to use?
Also, private healthcare is not the only thing people spend money on that can cause health problems. If your line of reasoning is to be consistent, then surely no-one who pays for anything that creates an unnecessary/avoidable health problem should get NHS healthcare. Okay. What if someone pays to go ice skating and breaks their ankle? Purchases a vehicle then has a road accident? Pays for cigarettes then gets lung cancer? What if someone buys booze and then hurts themselves while drunk? A staggering amount of accidents and incidents involve alcohol consumption. If you really want to save the NHS money, refusing treatment to anyone under the influence would be a good place to start.
I doubt you believe that anyone in any of the above circumstances shouldn't have access to NHS care. So, why do you feel people deserve NHS treatment when they have invested in activities that harm their health, but don't deserve NHS treatment when they have invested in protecting their health?