There are a few instances of teams having "songs" in the early 20th century - mostly notably "Blaydon Races" - but those were usually just local songs that were sung before matches rather than being chanted during the game.

The type of chants popular today really took off in Liverpool during the early sixties. There were a few contributing factors - the revitalization of the team that had been in the second division, the development of the music movement that would become Beatlemania, and the general irreverence of the period.

Edit: Here's a clip from Liverpool - Arsenal in 1964 that should be edifying.