You can read about Churching of Women in the Roman Catholic Church at
http://www.newadvent.org/cathen/03761a.htmBut I've never heard about "churching" in respect of a baby.
Derby Girl
The baptism service consists of two parts, the baptism itself and the public receiving of the infant before the congregation at a service in the church as a member of Christ’s flock. If a child is baptised privately, they ought to be brought to the Church as soon as possible to be received as members of " the flock of true Christian people". This is why the term "half baptised" is sometimes used, because the second part has not been carried out. However children that are baptized privately, are not half baptized, as it was commonly called, but "truly and validly baptized".
It is very rare for the C of E to hold private baptisms today, according to our vicar, and not encouraged.
Stan