The exchange between Churchill and Stalin I posted was meant to show that even though Churchill knew that the Soviet leader would not likely ease on Finland, he still tried to talk him into making peace with Finland by emphasizing the potential effect it would have on Germany and the Axis if the Finns were to drop out of the war.You can see from many biographies exmp. Mannerheims and from books considering finnish intelligence that Churchill and Mannerheim both shared the same oppinions considering communism. Churchill was sympathizing Finland during both wars. But the thing is that constant pressure from USSR and Stalin made Brittain declare war on Finland. He even said that declaring war on Finland was a mistake and the only military operation against finns was the bombing of the far northern port of Petsamo mainly occupied by german soldiers and ships. Mannerheim and Churchill sent a few secret letters to eachother considering the attitudes of allied nations toward Finland.
As I said before, while all of the Allies knew that the war in Finland was part of WWII, the Western Allies knew that they were a different case from Germany and her satellites and made efforts to take her out of the war.