The OP seems to divide the world into nationalized and "other". In practice, there is a whole continuum.
In France there are public and private hospitals, but they all issue bills (although generally the patient never sees them -- they go directly to the statei insurer, possibly with a small top-up from private cover for things like a better room); almost all doctors, pharmacies, and laboratories are private businesses, but 98% of their patients have health cards that pay between 70 and 90% of the cost up-front, and you have a co-pay that is mostly covered by top-up insurance, usually from your employer.
In Spain there is more state provision, so with standard national coverage (which almost everyone with a job has, via mandatory deductions) you will most often be seeing a medical professional who gets a state salary even in an outpatient context. But there is also a thriving private sector that is less well-connected with the state system than in France.
In the UK, an arms-length government department runs most of the hospitals and pays most of the doctors. As a resident, you never see a medical bill, apart from dental and pharmacy co-pays (for which people don't typically have insurance). There is also private sector provision, typically for elective procedures, mostly paid for with private insurance that is offered as a perk for middle-to-senior jobs. This is not especially expensive, because the private hospitals know that if you need something major (heart surgery or cancer treatment), you will be falling back on the state system.
Switzerland is quite close to the US: Almost everything is private and insurance-based. The key difference is that the government guarantees that you can get insurance. It's a bit like Obamacare without the complications.
And etc etc for most other countries in Europe, certainly in the old "Western" Europe, where many systems were put in place as part of post-WW2 social settlements.
