Located the OST of the film, music credits to Bob Adams / John Kentridge (also at 4:25 into the video).
It is hard to read the image of the back cover of the LP, but the track listing (1b of the Overture, and 6 on side 1) is "Kiss me mother" (trad British Army March) arr. B.Adams.
Some information about the 94th regiment, British army, whose marching tune this song is.
Unable to find a recording of it as a marching tune other than the film, but this sheet music Library of Congress via Picryl does look like it is the same tune.
George F Root was a composer of popular songs in a period spanning the American Civil War. He is only the arranger of the sheet music above - the tune has no other credit.
My idea is that the tune is of traditional (folk) origin, most likely Irish, since it appears in other versions in the bluegrass tradition, under the title "I am weary" which is also a lyric from the song. The origins of the tune could also be Scottish, given that the 94th Regiment who used it as a marching tune were a Scottish regiment. But again, they were based in Glasgow, which has a long history of immigration from Ireland. Probably never going to know the origins of the tune for sure. Anyhow,the bluegrass tradition has strong Scottish and Irish roots, since many of the people playing were from those origins at some point back.
This bluegrass version by the Cox family from the soundtrack of "Oh brother where art thou".
Another bluegrass version
by the Ralph Stanley and the Clinch Mountain Boys.
The liner notes for the Ralph Stanley 1977 disc credit the song to "Roberts". "Pete Roberts" is an alias for legendary bluegrass musician Pete van Kuykendall. He is credited under the name "Pete Roberts" as composer of the song at BMI with, reference here.
Whether PvK truly wrote the tune or was the composer of the definitive arrangement of an older tune - without him, that tune would be lost.