It didn't play in the background for me, but assuming that it was the one mentioned at the bottom of the page, it was The Mist-Covered Mountain, which I don't think is what we're looking for.  For an earlier discussion of both "boat" songs, not yet indexed by the search engine here, see  Canadian Boat Song

The only tune anyone came up with was the French one used by Thomas Moore.  If there was a tune given in the original publication of the "Scottish" song, nobody has mentioned it, so I'm assuming that it was set later on; I've heard it referred to any number of times over the years, but never sung.