I looked at the GRO index for Betsy's daughter Ellen's birth in Warrington and the mother's maiden name is Walker so not Betsy's mother, maybe a mother-in-law? The son Richard Barratt birth reg June 1/4 1848 has the mother's maiden name Walker,
Betsy is Elizabeth in 1861 and has a 5 year old daughter Jane Barratt but her birth reg does not give a mmn. Added:- this would normally indicate the mother's maiden name is the same as the child's last name but Betsy says her maiden name is Walker for the older children.