Not sure who is feeding you your "facts" but I live with both "nationalised" health care and private health care and if I need any medical treatment I usually get it immediately on private health but should the guy down the road who doesn't have private insurance have the same injury he will still get exactly the same care.
The only difference in treatment is the quality of the optional aspects of the case, for example I get private rooms, menu dining and all the bells and whistles and he doesn't.
Trust me having seen both your system and ours I would take ours any day not because our doctors or facilities are better because they arent but because our system gives everyone at least the basics in health care and those that can afford it have the luxuries of the private system.
We are more often treacherous through weakness than through calculation. ~Francois De La Rochefoucauld