I've recently read the claim that "La Cucaracha" was composed by an itinerant fiddler who accompanied Villa's army - The name Apolinario Gomez Garcia was mentioned. This is mentioned in a historical novel by John Harris titled "So Far From God" set in and around the Mexican Revolution and Civil War up to and into WW1.

Mind you, it shouldn't be taken too seriously, as the main characters are fictional with the political mess of the Mexican aspect of the Great War as the background to a pretty poor story, and so many of the supporting characters could also be fiction, but somehow, it doesn't read like it. Harris usually researches his books pretty thoroughly,

Anyway the whole concept of La Cucaracha being gradually adopted, first by Villa's own supporters as a kind of irreverent song about their hero, then by his co-belligerents as a mocking, less than reverent song, and finally by anybody at all who felt like singing, rings true. Much like any song started by any soldier anywhere really......