Extract from "The book of common prayer and administration of the sacraments, ..." published 1701 by the Church of England.
Where the Minister performing the follow up/second ceremony had performed the private ceremony
"Yet nonetheless, if the Child which is after this sort baptized (ie. privately), do afterwards live, it is expedient that it be brought into the Church, to the intent, that if the Minister of the same Parish did himself baptize the Child, the Congregation may be certified of the true form of Baptism, by him privately, before used: in which case he shall say this -
"I certifie you, that according to the true and prescribed Order of the Church, at such a time and at such a place, before diverse witnesses I baptized this Child."
If not the original Minister, he carries out something of an interrogation of those presenting the Child to determine if there was a previous ceremony.
"And if the minister shall find by the answers of such as bring the child, that all things were done as they ought to be; then shall not christen the child again, but shall receive him** as one of the flock of true church people, saying this
"I certifie you, that in this case all is well done, and according unto due order, concerning the baptism of this child, who being born in original sin, and in the wrath of God, is now by the laver of Regeneration in Baptism, received into the number of children and heirs of everlasting life. ...."
** - no "... or her!"