They hope they don't need it, but if you're getting operated on in an expensive private hospital and something goes badly wrong, or if you just fall down the stairs while you're there, guess what they do: they dial 999[1] and an ambulance comes round from the real hospital. And you won't be charged[2] for that service. Hooray for the NHS.[3]
[1] You can dial 112 instead if you prefer.
[2] Unless the private hospital charges you an admin fee for calling the ambulance for you, which I wouldn't put past those bastards. If it was profitable to slowly torture you to death that's exactly what they'd do to you. Personally I prefer to be treated by someone on a fixed salary who just wants to do the best they can with the resources available to them while earning the respect of their colleagues.
[3] This has turned into a bit of a rant, but I suppose it gives people a feel for the British NHS "religion".
The issue with British NHS religion is not that "the national healthcare shouldn't be free", it's that 95% of British population don't even want to discuss what's wrong with the NHS in the first place.
You were the first to have it and now you messed it up, compared to most of the european countries.
Believe me we know what's wrong with it, we're just careful to criticise it. The root of all those problems is a lack of cash. The current govt. is de-funding it to breaking point so they can bring in private contractors.
Of course, that private health insurance will generally dump them straight back onto the NHS as soon as they can describe an issue as “chronic” or it gets too complicated for them to want to pay for. Good quality healthcare plans (vs health insurance for acute events) doesn’t exist in the U.K.
Often it's the same doctor who will refer you to themself in the NHS after an initial private consultation if they work out that the insurance company isn't going to pay enough for your treatment to make it worthwhile.