The Copper family sing this one to a very English tune, and they pronounce it Cloddy. As far as I know, they think it refers to the Irish river Claudy. There's certainly an Irish tune too, from Donegal - quite different from the English one and also different from the Star-of-the-county-down melody that Eliza Carthy uses. They must surely have the Irish river in mind when they sing it there.

Which doesn't prove where it originated, only that it has English and Irish variants.