There are several tunes to which this song is sung, but I imagine you're talking about the one popularised by Planxty and, later, Paul Brady.  That tune is obviously related to Come All You Tramps and Hawkers (a relatively recent title), which was one of the tunes associated with [Blooming] Caroline of Edinburgh Town.  There are quite a few references to these, and to Lily of the West in the DT and Forum, but I'm far too tired to supply exhaustive links; you can find them all for yourself via the "Digitrad and Forum Search" on the main Forum page.  There's a particularly useful example at  The Max Hunter Folk Song Collection:

Caroline of Edingboro Town  As sung by Mr. C. W. Ingenthron in Walnut Shade, Missouri on November 19, 1958.  The resemblance is unmistakable.  The "original" melody probably has an even chance of being Scottish or Irish -it turns up fairly consistently in both countries, and has of course been found in England, too- but perhaps somebody else will be able to supply more precise details.

Malcolm