"Skiboo" (Skebaugh) harks back to at least the Sudan Campaign with Kitchener, in the song "Skiboo," ancestress of "Mademoiselle from Armentieres," or "Hinky Dinky." It probably is much older. The tune is ideal for marching with drums, and its origin seems to be lost.
Whether it has anything to do with "Usquebaugh" is problematical. My guess is that it is in imitation of a drum sequence.

Oh, landlord, have you a daughter fair,
Skiboo, skiboo,
Oh, landlord, have you a daughter fair,
Skiboo, skiboo,
Oh, landlord have you a daughter fair,
With lily-white arms and golden hair?
Skiboo, skiboo, skiboodley-boo, skidam, dam, dam.
From "Sound-Off!, Soldier Songs," 1929, E. A. Dolph, p. 82.

Also from "Tommy's Tunes," 1917, F. T. Nettleingham, p. 22-23:
Two German officers crossed the Rhine,
Skiboo, skiboo, and etc.
"A well-purged and diminutive version of a famous heirloom of the British Army; in its original state consists of about forty verses."

See threads on these soldier songs.

The 1870 sheet music of "Son of a Gambolier," at American Memory, is the oldest copy of this song that I have found, except for unverified statements such as those at the NOAA site. The composer (I would guess lyrics only) is given as "L. M.," Washington Rock, June 29, 1870. I haven't been able to identify him.