In the US, it's more of a planned/command economy.
A retail consumer gets the worst price. (Think the "rack rate that you never pay that's in the back of a hotel door) The government through Medicare/Medicaid/Tricare gets the best price. Everyone else is in the middle, on a spectrum of discount based mostly on size or some other market power.
In a single payer system, you end up controlling costs by compelling the providers to work for the system (similar to an HMO like Kaiser), prioritizing certain services and using regulation to discourage bad vendor behavior, like drug profiteering.
This is the basic premise of socialism. But it never works out. What you end up getting is either very low quality at high prices or very low availability.
Obamacare attempts to do exactly that, and as a result the providers are simply leaving the "market". Meanwhile, price has skyrocketed, and quality of coverage has dropped dramatically.
My marketplace plan would likely bankrupt me if I got a serious condition, however the coverage I had before Obamacare was very good and very cheap. I'm not that much older!
The NHS has its flaws (having to wait for non-urgent or non life-threatening issues to be treated, but they are in fact treated, just at a lower priority) but if you present at the hospital with chest pains you'll be sorted out straight away, without having to pay up front - free at the source of treatment, one of the founding tenets of the service.
It also doesn't preclude a private system that works alongside it, which many companies offer to their employees. This allows you to get seen quicker for the non-urgent stuff - if it's life threatening you'll be taken straight to an NHS hospital, but have your follow up care privately.
We pay less of our GDP on healthcare, yet have comparable, if not better, outcomes.
With the current "privatisation" that's occurring, vendors are invited to bid for the work, no compelling, it's there if they want it. If nobody comes forward to provide it, the NHS will do it instead.
Obamacare is actually working very well in compelling the providers to participate. In most regions, private practices are rolling up into regional health systems, often around anchor hospitals.
That's a very commonly held perception. However it's not the case.