A single payer NHS actually does change the overall cost of healthcare: the UK pays about half as much per person as the US does for its healthcare system as a whole with no noticeable difference in measured effects.
The reason is simply market forces: people import drugs to the US from Canada because the power of single payer there changes what pharmaceutical companies can charge Canadians for drugs. The UK pays less for drugs, doctors, the whole shebang. Operations are cheaper in the UK and Canada (not to mention the rest of the world, which is more extreme in general), period.
The takehome point is simple: pretending that consumers must be price-takers for any good as vital as healthcare entitles producers to arbitrary profit. I hope not to spend all my savings on healthcare and my best hope is to get some kind of market power, be that from an insurance company or the government.