The Captain and His Whiskers (Roud 2735) is known in English and Scottish tradition; certainly not American and Irish only. They Marched Through the Town is an English song in origin; Nathaniel Thomas Haynes Bayly (1797-1839) also wrote the words to The Mistletoe Bough (and O! No, We Never Mention Her, among many more), and the equally prolific Sidney (American songsheets sometimes call him Samuel) Nelson composed the music for The Rose of Allandale.

The British Library catalogue lists the following:

[1] They marched through the Town. Listen dear Fanny. Away to the Mountain's brow. Marriage of the Rose. [Songs.]
TOWN
[London, 1845?] s. sh. 4o.
[2] As they marched through the town. The Captain and his Whiskers. A comic song, etc
1869
[3] As they marched through the town, marche militaire ... pour le Piano
Mattini. F
1868
[4] O they marched through the town. The Captain's Sly Glance. [Song.] Poetry by W. J. Wetmore and T. H. Bayly
1864
[5] The Captain, or Oh! they marched through the town. [Song.] (Arranged by W. A. Owen. Written by T. H. Bayly.)
Owen. W. Alonzo
1861

This would suggest that the "Whiskers" development was quite new at the time of the Civil War, and the original song not much more than thirty or forty years older.