It is good to know that in your case (TTT) the outcome is satisfactory, congratulations and best wishes for a speedy and complete recovery.
As for the NHS, of course anyone getting any attention when in need of medical help will be grateful. The separation of the twins demonstrates what the individuals and facilities are capable of. The best of these, as for example in the USA and elsewhere, are the best worldwide. It is of course reasonable to assume individuals working within the NHS always do their best and we should treat the too many examples of awful blunders, ill treatment, murder, etc. within the system as being "in spite of best efforts".
TTT is correct that the NHS is among the examples where nobody is asked for payment at the outset but worldwide such examples are very many, particularly in emergency situations (the USA horror stories are legend and most of them true but not necessarily representative even there). But the NHS very quickly ascertains whether the person is entitled to treatment no-questions-asked or not, particularly in non-emergency situations. If reception is not able to establish this then the same obstacles show up as they do elsewhere - another healthcare myth is that you can get NHS treatment and walk off no-questions asked. Non-emergency has the same ring-fencing as in most other places - paperwork and questions aplenty.
Try going to any NHS medical facility, introducing yourself as John Smith and expecting to be seen without "going through the paperwork" - you will be met with questions that identify you as registered with the NHS or not. That is exactly the same scenario at the vast majority of (all?) receptions to a national/state health service anywhere - at the majority of these all emergency treatment is entered into as a matter of urgency and questions asked alongside and/or afterwards.
I do not have universal knowledge but, in those countries I am familiar with, as soon as you become resident you become eligible for treatment under the state/national health service (and you must pay tax), literally instantly - except for the UK where you do not reach that status until three months after you are answerable for paying tax (that of course funds it). I think I am correct in saying that you remain resident in these countries (that I am referring to) until such time as you declare yourself non-resident. Until then you remain accountable for tax and eligible for healthcare. In the UK, resident or not (and the UK does not have a useful system for establishing that) you must not be outside the country more than a certain number of days or you lose entitlement to healthcare - but you remain accountable for payment of tax (NI contributions are a tax).
No, although at its inauguration it was almost certainly exemplary, today the NHS and everything that surrounds it is no longer that. Sadly, internationally it is somewhat mediocre.
In an international survey published in The Lancet the NHS was 23rd in terms of quality, efficiency, accessibility, cost, etc. among way over 100 different countries analysed. Thus, it is not correct to say that the NHS is the envy of the world, although that is clearly the (mistaken) perception in the UK. At least 22 countries will be happy that they have a better health service than the UK does. It seems a logical conclusion that, for the patient, getting appropriate treatment is uppermost, even if some payment must be made. Not being charged in cash for inferior treatment (medically or otherwise, including the cost in tax) is not necessarily, or even likely to be, the first choice for the patient and his/her family.