Thanks, Masato- the comments by Levy put a much better time frame on the song. If "Lady Jinks" came on the scene in 1868, then the Captain was around before then, as the Glasgow broadside of 1866 indicated.
I am still curious about the Maine Coon cat supposedly named Captain Jinks in 1861. If true, this would place the song at the start of or just before the Civil War, and would verify the statements that the song was "popular during the Civil War."

A number of websites, including Bluegrass Messengers, call the song American, but there is no evidence to support this that I can find.

A UK site says the words are by a T. Maclagan and song by Lingard, for the British Lingard Company in 1868 performing at a theatre in New York; "it was sung by a chorus of girls in military costume" (this may help explain the Dame Edna get-up Lingard is shown wearing on the cover of the 1868 sheet music). This from a book by Donald Clarke that may or may not be well-researched. Certainly the song was performed before 1868.
(http://www.musicweb.uk.net/RiseandFall/three.htm)