Various notes from the older carol collections can be seen at Hymns and Carols of Christmas: I Saw Three Ships. The link may not work in some older browsers due to the way stylesheets have been implemented.

It certainly appears that the "Forbes text" (frequently referred to on websites, but rarely with specifics; presumably all copied from a single source and not checked) is that quoted by Ritson and Elliott/Shire. John Forbes, Cantus: songs and fancies, to severall musicall parts... (etc), (Aberdeen, edition of 1666) is the earliest certain appearance in print; it should be noted that the three distinct editions of the book (1662, 1666 and 1682) contained a mix of Scottish, English and Italian material.

The tune in Forbes is presumably unrelated to those we are now familiar with; examples found in oral currency have tended to be sung to variants of popular tunes of the day such as Nancy Dawson or Green Sleeves. The song was widely published on broadsides during the 19th century, as mentioned earlier. Examples can be seen at   Bodleian Library Broadside Ballads:

As I sat on a Sunny Bank