My father's had severe (and unusual) kidney issues and has been treated by both NHS and private. Going by how he was treated, I'd say private all the way. The NHS have done nothing except screw up, and at one point nearly kill him (sent him home after a particular procedure and he started bleeding internally). Going back a ways, when he was a younger man, they also failed to diagnose the fact that he'd managed to break *all* his ribs, right next to the spine after a car crash. They told him he was fine and sent him home. It was only when he came back complaining of back pain that they x-rayed him and found out it was a miracle he hadn't been permanently paralyzed.
He also managed to contract a new strain of meningitus back in the 90's in Africa, and luckily was treated privately in S. Africa. They had to create an entirely new drug to combat the illness, but because he was private, the insurance paid up. If it had been NHS they would have tried for a few weeks and then just given up (considered it a waste of resources).