The Churching of Women was the common name for the
Thanksgiving of Women After Child-Birth. This site gives a comprehensive explanation of 'Churching'
http://users.ox.ac.uk/~mikef/church.html#introAll the Book of Common Prayer says is "The woman, at the usual time after her delivery shall come into church......" It says nothing about the mother not being at the baptism.
In Jewish Law the period of uncleaness, which came from the bleeding, not from the birth was, after the birth of a baby boy, 40 days, half the period of for a girl, 80 days. After 40 days(or 80days) the time of purification was completed, and the woman went to the priest with a sacrificial offering after which she was ceremonially clean from her flow of blood.
Stan