This is the Scottish version of "Bye, Baby Bunting"...

Hushie ba, Burdie-beeton - Scottish Children's Songs - Scotland - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Notes

(1) Burdie-beeton is a fondling term for a little child as per the site, Dictionary of the Scots Language.
(2) Lammie/lambie is a term of endearment and a diminutive.

Thanks and Acknowledgements

The illustration is from The Only True Mother Goose Melodies (published and copyrighted in Boston in 1833 by Munroe & Francis).