It all gets paid by somebody. For everything that goes through a normal health insurance policy, about 20% goes to the company as profit. If you nationalize it, Medicare allows 5% as overhead to those same companies to process payments. So we collectively would pay 15% less for every medical bill now covered by a normal policy.
That's the big picture. Of course there are a huge mess of subsidies, government grants to hospitals, CHIP, Medicaid, and so forth. Almost nobody paying what it actually costs, so there are a lot of toes to get stepped on. But costs would not go up.