LOL... Britain doesn't have single-payer.
They have an NHS, which is a government-funded hospital system, but they also have a private system, too. That's not single payer. Single payer is allowing one entity, usually the government, to monopolize the insurance system so that only that entity can buy health care. Canada's provinces are single-payer entities.
So, you want or need to have a procedure done, but the "single payer" won't cover it? That means you can't get it!
THAT's single payer. Believe me, nobody truly wants it.