F. J. Child's introduction to #10 is full of international examples from folk tales, of the corpse reconstituted as musical instrument or other everyday object, and going on to reveal truth or treachery. The Grimm's fairy tale 'The Singing Bone' is one of them.

Going back to the actual ballad, the question posed by the OP is an interesting one. There seem to be two main strains for Child #10: one with the alternate line refrains 'Bow Down, Bow Down / Bow and Balance (or the Bough it Bends) to Me", and the other the 'Binnorie / by the Bonny Mill Dams of Binnorie' type (the 'Edinboro / Bonny Saint Johnstone Stands Upon Tay' type is very similar, textually).

The first strain is the commonest in North American collections of the 20th century (particularly in the Appalachians) and has also been collected in England; in this form it almost always concludes with the miller either finishing off or rescuing the drowning sister, with no supernatural element. The motive for the murder, originally jealousy over a lover, is here changed to a quarrel over a hat (what's the world coming to, eh?).

The second seems restricted to Scotland apart from one Northumbrian example; this strain (when collected in a reasonably complete form) usually does include the magic harp or fiddle.

It would be tempting to propose from that evidence that an 'English' strain, with the 'Bow Down' refrain and a simple murder-ballad theme, was the ancestor of the numerous Appalachian variants, while a separate, supernatural Scots strain existed but never made the crossing. But of course nothing is ever that easy. The oldest example listed by Child is an English broadside from 1656 with the refrain 'Hie downe downe a downe-a', which includes very detailed references to the body parts used in building the fiddle (the nose becomes the bridge, and the veins the strings, etc.). Meanwhile, Appalachian versions exist that do include an account of the magical fiddle, but these have different refrain types: either 'Oh the Wind and Rain' (fairly rare in tradition though popular currently), or 'Jenny Flower Gentle and Rosemary', which - confusingly - has associations with at least three other Child ballads.

It's quite possible that entirely different versions of the ballad might have arisen independently from the same raw material (folk tale, minstrel romance, whatever). However, in this case it is possible to compare the texts of the two main strains and find certain similarites:

The stanza from the 1656 broadside 'Somtymes she sanke, somtymes she swam / until she came unto the mill dam', survived in recognizable form in several Scots 'Binnorie' variants collected nearly two hundred years later, but also in 'Bow Down' versions found later still in England and the US.

Another common sequence, in which the drowning sister fruitlessly offers inducements in return for rescue ('Sister, sister, give me your hand / and you'll be the heir to all my land') did not appear in the 1656 broadside, but occurs in several of the 19th century Scots versions listed by Child, and carries on turning up in English and N. American versions right through to Derbyshire singer George Fradley's 1980s recording for Veteran.

Such textual similarities would argue against two versions of the ballad having arisen completely independently. But we know so little about the composition of the ballads that any conclusions are going to be speculative. Did some 18th or 19th century hack compose the 'Bow Down' murder ballad, cannibalising verses from an earlier broadside? We just don't know. It's at times like this that I miss Malcolm Douglas!