I think that the "Lady Franklin" tune is a minor variant of the "Croppy Boy" tune.

As 1798 songs go, Croppy Boy seems to be quite early, James Catnach was printing it on Broadsides between 1813 and 1838.

John McCormack recorded the song on a wax cylinder in 1906 and credited it to William B. McBurney. The tune is noted here as being originally ...

"Cailin o cois tSiure me" (I am a girl from beside the river Suir). According to Nicholas Carolan, this is "the oldest dateable Irish melody" (16th century).

Hope this is useful.

Cheers!
Ian