“Defending the free world” doesn’t require Christianity. Some Hindus, some Buddhists, some Moslems, lots of Jews found that their interests coincided with those of “the West” in World War II.
I’m not saying that Churchill wasn’t a Christian, but he wasn’t any kind of doctrinaire denominationalist, and he wasn’t a steady churchgoer. He wrote his own ticket, as they say ... and certainly wouldn’t have conflicted with Tolkien on strictly Catholic vs. Church of England grounds.
If it’s not Christianity, then it’s Judaism. Stalinism had no interest in defending the free world, only defending itself, and had to lean on the Christians, to its chagrin.
Undoubtedly there were lots of churchgoers in the political/ideologocial sphere that opposed Churchill, I daresay, and Churchill’s struggles with many personal problems (including depression) have no shortage of documentation; but it was Churchill that showed faith ultimately.
I don’t believe Churchill would have gone against Tolkien on such grounds either. But it’s most likely that Tolkien did see Britain in terms of “Mary’s Dowry” either way.