Interestingly, it seems that Annan Water has only been published with music once, in the early 19th century; I don't know what that melody was, or whether it was traditionally associated with the text. So far as I know, everybody nowadays (including, I'm told, Niamh Parsons) uses the tune that Nic Jones set to the song (not always crediting him), which he found in the Journal of the Folk Song Society, vol.I no.2, 1900. It was collected by Lucy Broadwood in Surrey, and belonged to an English song called The Brisk Young Lively Lad. Nic changed it a bit to suit his anglicisation of the text from Child, which in turn came from Scott's Minstrelsy of the Scottish Border.