Consider for example, your BUPA insurance or whatever, that allows you to see a specialist of your choice within a few weeks. Why can you see them within a few weeks? Because other people don't have BUPA, don't want to or can't afford to spend privately, and so they wait much longer to be seen, probably by the same person, on the NHS at no personal cost.
Consider, then, if publicly funded healthcare were to cease to exist altogether. So that that option which has kept people off the private waiting lists, ceases to exist. There are still the same number of specialists, if nothing else changes - if the training bottlenecks and poor incentives remain in place - there will be no additional specialists. If the bottlenecks and incentives change, it still takes 10-15 years plus to train more (and also begs the question why the training bottlenecks and incentives can't just be improved within a publicly funded system).
So either:
A} You continue to pay for private health insurance or upfront for healthcare, but so does everybody else and it no longer gets you seen within a short time frame or gives you access to higher quality than average services. You are all now paying to wait around as long for a similar quality of care that used to be available in a cheaper, publicly funded system.
B) A competitive marketplace develops between insurers / providers, such that you can still be eg seen within a short time frame but you'll have to pay a hell of a lot more for it, seeing as healthcare at no personal cost is no longer part of the marketplace. Receiving high quality care on an urgent timescale becomes the domain of the super rich, and/or ordinary people have to bankrupt themselves to access it. Ordinary people have to limit the healthcare they receive to what is personally affordable.
C) Other people don't compete with you to be seen because they simply can't afford to do so. So maybe your waiting list doesn't increase as much but this is because large sections of society don't access healthcare at all.
D) Public, free at the point of care healthcare does exist in some capacity; but it is limited or means tested and/or delivers a poorer service. So that if you are lower income you can access second tier care, if you are middle or higher income, you can choose to pay a consistently high price in insurance or take on the risk of personal liability for healthcare costs - whilst accessing a higher tier healthcare that isn't much better, and is very possibly worse in terms of a higher proportion of treatments being inaccessible on cost grounds, to the previously existing publicly funded healthcare. And is delivered by the same people.
Or some combination of the above.
Does any of this sound maybe, familiar?