My mother always sang that to us, but the words were a bit different.

She always sang:

Baby owlet, poor little owlet
Singing as the sun doth shine (repeat)

Won't you send me your sweet pinion (and then she'd sing another verse with "won't you send me your sweet kisses)

It was our favourite lullaby when we were young and I always sang it to the babies when I was a nanny.