The original question was about songs which are assumed to be American, but "which have Scottish roots", a point which most contributors so far seem to have ignored.  I fear that, knowing little about what is assumed in America to be American, I can only add this postscript to Art's earlier comment:

Though I don't doubt that Dumbarton's Drums was known in Ireland, it seems to have been known rather earlier in England (Playford's Apollo's Banquet for the Treble Violin, 1690) and earlier still in Scotland (Skene MS., c. ?1615-?1630, though possibly later); there seems to be no question but that it originated in Scotland.