Interesting speculations. Little evidence seems to be available, but the search may be productive.

While checking on British WW1 songs that persisted into WW2, I ran across an item (sorry, can't re-find) that "Skiboo" had been published in a local army paper in India about 1910. The suggestion was that it did date back to the early days of the Empire, or around 1800, but no more information showed up.

"Johnny I Hardly Knew Ye" has been suggested as a precursor, but all records seem to be post-WJCMH.

I would not be surprised to find that the tune to WJCMH has European origins. Nothing particularly Irish about it. "Original traditional tune" preceding all of the lyrics- seems logical, and the rhythm is perfect for marching with just drums and perhaps fife.

"See the elephant" precedes the Gold Rush of 1850. Lighter, "Historical Dictionary of American Slang," quotes it from 1835, equivalent to seeing the sights; it was also used in Kendall's "Santa Fe Expedition," 1842-1844.
(Wonder if vol. 3 of this useful compendium will ever be published).