Yes - it does rather depend on what the character's own background is, as lullabies tend to be passed down within a family. If, for instance, the singer was from a Welsh family, they might well still sing it in Welsh. I doubt if they would have gone over to any English words for that particular one. It might have happened with All Through the Night because there were very popular English versions of it around, but Suo Gan wouldn't be in that category.

If the singer is less culture-specific, maybe something like the, "Mama's gonna buy you a mockingbird" one would suit ... although I'm talking through my hat here; I don't know anything about its background or age.

sian