This guy uses the "Son of a Gambolier" tune:

http://www.youtube.com/watch?v=ERyLNa-WBQU

Here's Ry Cooder singing it to "Joe Bowers":

http://www.youtube.com/watch?v=bEvBCnHPM4Y

I knew a CW re-enactor back in grad school (1975) who sang the song to "The Yellow Rose of Texas" as well as to "Son of a Gambolier."

"Joe Bowers" is rather similar to the the usual tune of "The Lily of the West."