Belated reply to SteveG:

It's possible there was an indirect tie between the now-well-known "Babes in the Woods" and the older, lengthy ballad that was still known in Gardiner's time, but it is one of common subject rather than textual or musical derivation. If you make up a new song about Robin Hood, borrowing elements from the common lore, can anyone realistically assert it dates back to the 14th century?

We know who wrote the newer "Babes in the Wood" song and when, and we have sufficient versions of the older ballad to show they have at most only an "inspired by" connection—they share no metric structure, tune or common lines. The newer song was created essentially from scratch in the 1790s.

Apart from the clay figurines, it's unclear what Gardiner's friend may have used as the basis for his lyrics. As I mentioned before, the song follows the barest of story lines: they were "stolen away" (miscreant undescribed) and "left in the wood"; they cried, they died; then a robin covered them with leaves and sang. The prologue says that "mother related the story to me", which suggests that the lyricist's source was a nursery tale rather than a ballad. Over time there have been multiple legends of "babes in the wood": it was such a simple method to dispose of inconvenient spawn. (Don't tell Prince Andrew.) No sure link between the song and the once-popular ballad, much less any particular legend, has been established. This song's "long" history effectively dead ends in the 1790s; anything posited before that is pure conjecture.