The song is not “Glory Hallelujah”. It’s Battle Hymn of the Republic. Probably not PC enough for 21st century America.
A slightly reworded version is the marching/jumping song of American paratroopers as well. In that iteration, it's known as Blood on the Risers. And it's CERTAINLY not PC enough for 21st Century Americants.
Song begins at about 1 minute into the clip from Band of Brothers.