My understanding is that Churchill was not exactly a popular prime minister during the war, but the war coalition gave the feeling to the British people there was not much they could do about it. When they got the democrat chance to get rid of him, they took it.

Later the popularity of Churchill was put into British myth.

However, this article probably has a lot of truth to it. There is no doubt, from our historical hindsight, that Churchill was a great war-time PM.