I had the Barry recording only briefly from the library in order to get the text, so my memory of the tune is vague.  As Brían says (thanks for the jog to my memory), it does seem to be a modified form of Tramps and Hawkers, which is a recent name for it; it's carried a lot of songs in the past, notably Caroline of Edinburgh Town, by which name it's often prescribed as melody for broadsides.

It later became attached to a song called Paddy West; there is a midi with the DT file:  Click to Play.

Is that at all close to the tune you have in mind?  Margaret sang it differently, of course, but it might possibly help you for purposes of comparison.  In other modified forms, that tune also carries some versions of Lily of the West and The Lakes of Ponchartrain.

Malcolm