There are actually two versions of this cartoon. The first, "Peace on Earth," is from 1939 and was directed by Hugh Harman for MGM. It was nominated for an Academy Award for Short Subjects (Cartoon) but didn't win, losing out to Walt Disney's "The Ugly Duckling." A few claims have been made that the short was actually nominated for a Nobel Prize (such a claim was made in Harman's obituary) but has never been otherwise substantiated, and isn't listed in the official records for Nobel nominees. This version has a grandfather squirrel telling his two grandchildren about the demise of humanity (the young squirrel's being prompted to ask by the lyric "goodwill towards men," given that they didn't know what "men" were). The combat depicted is clearly inspired by WWI. The second version was made in 1955 by Hanna and Barbera, again for MGM, under the title "Good Will to Men." Hanna and Barbera's version had a mouse Deacon telling the short's story to his choir of young mice, had more overt religous references, and, from the combat depicted, was clearly referencing WWII.
Cartoon Network has shown "Peace on Earth" a few times, usually around Christmas. I remember seeing it as a child and, like you, being rather haunted by it (never seen "Good Will to Men" myself). It's good stuff. Hugh Harman and his partner, Rudy Ising, tend not to get the recognition they deserve. They made some great cartoons.