It is indeed from a British Broadside. You can see examples at the Bodleian Library's broadsides site.

Interestingly, most versions say "The Banks of Sweet Dundee," not "THE sweet Dundee," so we don't have to assume that Dundee is the name of the river. Given that, it could well be set near Dundee on the Tay.

That's what the National Library of Scotland has decided...you can see a version of the broadside, and their comments, here.

There was also a sequel ballad, in which William returns home from sea and tests her loyalty by disguising himself and telling her "I knew William and saw him die," and watching her go mad, then saying "just fooling, I am William!" Typical broadside stuff, and they live happily ever after. You can search at the Bodleian for "Answer to Undaunted Mary" to find that one.