The entry at the Traditional Ballad Index is a little misleading, in that it only cites the Roud number for Dear Companion (411). There are far more extensive entries under Roud 459 (Go and Leave Me and other titles). The song seems to have been known throughout Britain and Ireland at the end of the 19th century, and of course was widespread in the USA and Canada. There is an undated broadside example from the Firth Collection at the Bodleian Library website, but no image is available at present.

It must have been widely published, and with music, too, at some point, as other songs were set to its tune; but I don't know where you might easily find an example of the song as written, nor whether it is more likely to have originated in America or Britain. There appear to have been two songs; the original and a sequel ("answer to"): the Roud Broadside Index refers to Harding's Dublin Songster part 5 (c.1900) which includes Go and Leave Me if You Wish It ("It is years since first he met me...") and Go and Leave Me if You Wish It (Answer to) ("Years have passed since last we parted...").