The Mouse on the Moon (1963)

The Tiny Country of Grand Fenwick has a hot water problem in the castle. To get the money necessary to put in a new set of plumbing, they request foreign aid from the U.S. for Space Research. The Russians then send aid as well to show that they too are for the internationalization of space. While the grand Duke is dreaming of hot baths, their one scientist is slapping together a rocket. The U.S. and Soviets get wind of the impending launch and try and beat them to the moon.