UK you pay NI (National Insurance). This is a set % so effectively a sliding scale based on earnings. Nothing is perfect, and funding comes from elsewhere too and picking the bones out to see exactly where and how effectively taxes are used - see one article as just one snap shot entry in to a very long and ongoing issue. EDIT: Note there is no need to be in employment to receive the benefit, but if you are taxes apply yes. For added clarity everyone is fully insured if you like.
After WW2 the Welfare State was set up - free health, free schooling and everyone gets a house (latter if can't afford to buy which was and is the real goal)..say free meaning at point of delivery, save for housing which was to be affordable. All very good but to say there are not problems (as with anything in fairness) is way off too. America is a fantastic place but problems exist same as anywhere, and this probably sums up my view Obamacare - said as may also help explain some o the differences I perceive at any rate v the NHS (realising the comparison was not your ? but anyway...).