The Country Girl's Policy [W and C Dicey, Bow-Church-Yard, London: Douce Ballads 3(17b)] rather belongs with Bung Your Eye / Quare Bungle Rye than The Basket of Eggs or Brisk Young Butcher / Christmas Goose, I'd have thought, having much the same story (baby palmed off on complete stranger). Bite Upon Bite [also Dicey: Douce Ballads 3(4a)] is closer though more complex; I wouldn't know if there's any direct connection.

Roy Palmer (Bushes and Briars, Llanerch, 1999) mentions a printed set of Basket of Eggs which appeared in The Man of War's Garland of 1796 as Eggs and Bacon; this appears to be the earliest known example of the song proper. There is a copy of the Garland in the Harding collection at the Bodleian (Chapbooks A. 15, no. 19. Wehse 196).