Carol, the Welsh Folk Song Society records:

"Noted by Mrs.Mary Davies of Whitchurch, Cardiff, Sept. 1908, from the singing of Mrs. David Evans, who had heard it sung at Llangammach Wells, in 1828, by Thomas Elias (Y Bardd Coch), a travelling taylor." (J.W.F.S.S.,Vol. 1, Part 2 (1910), No.7).

The Dictionary of Welsh Biography has a short entry on him but doesn't credit him with the authorship either. And the verses are pretty consistent with the overall body of 'hen benillion' (old verses: folk poetry). I suspect that the family history has reworked 'singer of' to 'composer of' ... which is common enough. But if the family has any details to add to the debate, it would certainly be interesting to hear them and follow them up. Always nice to gather in any available info.

sian