I used to have a comprehensive medical insurance as part of my employment in the healthcare industry, so have experienced both, in the UK and abroad.
There is no private A&E in the UK effectively. A&E is a service you need close by, whatever location you are in when you get sick. With the horrendous congestion in cities, a solitary private A&E facility can be inaccessible even to close by locations in an emergency.
You can’t always predict what will happen to your health as you age. Some people think that they will simply buy their way out of trouble, or that paying automatically means that the care or outcomes will be better, but it isn’t the case. There are generally higher clinical standards in the NHS, especially when the staff are not over stressed and under resourced. This is one of the reasons why former NHS staff are so valued in other countries.
If your case is more unusual or challenging (and you can’t always know this in advance), or you need access to the latest clinical trial drugs, you are nearly always better with an NHS department in a teaching hospital. I learned this the hard way by wasting four years on private consultations, paid for by insurance. If I’d toughed out the 14 month wait on the NHS rather than going private initially I would have been fixed much faster.
Finally, private hospitals where I am are struggling to recruit staff into all sorts of positions. Nursing care, post operative, was chaotic and generally very poor when a friend recently paid for an elective procedure, and the rooms were not clean enough.