Neither Irish nor Scottish in origin, this song and its relatives have been most often found in tradition in the South of England and in Wales, though examples have also been recorded from Scotland, Ireland, Canada and the USA; there are also close analogues found in France, Britanny, Switzerland and Denmark.  Peter Kennedy (Folksongs of Britain and Ireland, 1975) was inclined to think the Welsh and Breton versions the oldest, but others would perhaps disagree.  It may perhaps be that the rattling bog part was introduced in Ireland; it seems that most traditional versions refer instead to green grass or green leaves.

The tune commonly associated with Rattlin' Bog is used in Northwest England for the Morris, and in Ireland for the Polka; whether originally English, Scottish, Welsh or Irish is hard to say, as there are resemblances to many other tunes from all four countries.  I suspect that it's a descendant of Green Grow the Rashes, which would make the tune Scottish originally, but that's only a guess.

Malcolm