Churchill may have been a great man, but he was also a politician. The pressure for an NHS came largely from the Labour Party with power was shared in the national government during WWII.
You don’t get into high office without having a certain sense of pragmatism. Apart from having to make concessions to his Labour colleagues, he knew very well that the public mood at that time supported universal healthcare, and resisting it would have been political suicide in a post war world.
As it happened, even that wasn’t enough for Churchill, and he was voted out in 1945 by a public who didn’t generally share his Conservative convictions.
They were also ready to shuck the British Empire, which Churchill steadfastly refused to do.