Although large parts of both Saro and Bunclody are composed of "floating" verses, the quite close parallel certainly would seem to suggest a connection; both songs being descended from a common ancestor, perhaps. There are some broadside examples of the second half of the 19th century at Bodleian Library Broadside Ballads; only one, an issue by Brereton ("The World's Worst Printer") is legible, though, and that barely. However, it's virtually identical to the text given by Colm O Lochlainn in his Irish Street Ballads, which he learned from his father, who would have been around 10 years old when the Brereton sheet was issued.

The maid of Bonclody Printed c.1867 by P. Brereton, 56 Cook Street, Dublin.

Lochlainn's tune for Bunclody doesn't much resemble the best-known Saro tune, but one that does is the melody used by A.P. Graves for his song My Love's an Arbutus. As Bruce Olson pointed out in the thread where that was posted, the tune is from Stanford-Petrie (no. 507), where it is called I rise in the morning with my heart full of woe, or, The Coola Shore. At this point I grasp at a tenous connection; in Sam Henry's Songs of the People, there is a song, If I Were a Fisher, which begins When I rise in the morning, to my garden I'll go... which, like the song that follows it, The Star of Benbradden, is pretty clearly a member of the Saro / Bunclody song-family. I don't think it's only my imagination that detects a kinship between the Fisher tune and that of I rise up, either.

Of course, the floating verses mostly turn up in English songs, too, but given the structure of the text and these apparent tune correspondences, in this case the strongest argument seems to be for placing Saro as an Irish song changed in some, relatively minor, particulars to suit its new life in America.