Is it English or Scottish?  So far, Tamara, we haven't been able to establish that it isn't American!  It certainly is one of the most-parodied songs you could hope to find, though these parodies all seem to have originated no earlier than the turn of the century, and I'm guessing (without documented evidence for or against) that song and tune both came to the UK from America.  Until somebody finds real evidence of the song -Barney or Bonny- existing anywhere at all before the 1870s, I think that we have to assume that.  Incidentally, Cecilia Costello's My Johnny, which Bruce mentions earlier (Leader Records LEA 4054), doesn't appear on the track-listing in the Leader Records Discography at  Leader Records Discography  (Please tell them if they're wrong, Bruce!  They'll be glad to know.)   There is a recording available on one of Peter Kennedy's Folktracks cassettes, though: The Greenwood Side-I-O (Cecilia Costello and Sam Bennett) where it is called My Barney (lies over the ocean)....

Malcolm