Various versions of the song, most notably by James Hogg (Jacobite Relics, second series, 1821; transcription above) and Allan Cunningham (Songs of Scotland, III, 1825), appeared following its publication in the Scots Musical Museum (II, 1788, number 187). The SMM version was contributed by Burns, and consisted of three verses only. The song is a perennial in anthologies of Scottish popular song (and in British and American C19 songsters); for a couple of broadside editions at  Bodleian Library Broadside Ballads, see  O'er the Water to Charlie.

There are also other broadside songs set to the tune, which turns up frequently on its own in C19 MS tunebooks throughout the UK. For historical notes by the late Bruce Olson, with ABCs, see thread  Tune Req: Over the Water to Charlie  (link also given above).

Further detail is in Frank Kidson's article from Grove's Dictionary of Music and Musicians (1910: appendix) at the  Musical Traditions website:

Over the Water to Charlie.

Of the two DT entries,  Over the Water to Charlie was transcribed by ear (presumably) from a record by 'Golden Ring'; their source is not mentioned.

Over the Water is the Scots Musical Museum text. The DT contributor, having presumably got it from an anthology, attributes it to Robert Burns; but this is misleading. Kidson considered it an older song 'enlisted in the cause from the accidental use of the name "Charlie"', and went on to mention that it had been '''cried down" at Edinburgh Cross and other places, that is, officially prohibited, by means of the town-crier, from being sung during the Jacobite rising.'