A "rout", in the sense that it's used in this song, is a muster of a regiment - a calling together of the ranks ready for some activity of other. It was a common experience, and the Blues (latterly Blues & Royals) was a popular regiment.

You can read in Jane Austen's "Pride and Prejudice" how the girls of Meriton were sad when the militia upped stakes and moved on. So the "lassies crying and tearing their hair" was probably quite common!