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.
(3) "Boukie" is an endearing term and a diminutive of "bouk" or "bulk", meaning person. You can also find the variant "boulie" in the place of "boukie". According to "A Glossary of North Country Words" by John Trotter Brockett, "boulie" is an obsolete word for "beloved".

Comments

There are longer versions of this lullaby.

Thanks and Acknowledgements

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