Unfortunately, the sad truth is that UK can no longer afford to give free care at the point of need to all.
Firstly, those not entitled, should pay - eg foreign "health" tourists - which would help. It's v easy to set up; not embarrassing, not offensive - the patient either has a UK passport + Nat Ins no., or they do not.
It's not hard to organise. My experience when my OH had an epileptic seizure (two days after being at high altitude) when we were in Ecuador - my wife and I were taken by blue light ambulance to a not far away modern hospital. My wife was taken straight in to be examined by an urgent care dept med team, and at the same time I was taken straight to the main reception desk. I had to give our details etc for registration, and as I nor my wife are Ecuadorian nationals, I had to give my credit card impression. Thus after the tests etc, when my wife was declared OK and medically fit to be discharged, I was handed the receipted invoice taken from my credit card. It was all v simple and efficient. Why can't we do this in UK? I wasn't offended, nor did I feel "excluded".