Every single hospital, clinic, dispensary, consult, pharmacy, etc will have multiple-squares-long queues of people who has been letting their problems fester for not being able to pay it.
It will become obvious that the medical infraestructure is deficient.
And of course, libs will froth in rage. Chuds too but who cares about them.
More likely is they wouldn't get rid of an NHS-like system wholesale in a short period, but they would do what they always do with public services and slowly kill it through chronic underfunding until everyone hates it or at least isn't satisfied with it. Just like the USPS, just like public transportation, just like Social Security.