Baptism does not require a priest. All it requires is the intention and the words given in the Bible: “I baptize you [name], in the name of the Father, and of the Son, and of the Holy Ghost [or the Holy Spirit].” Particularly in an emergency, or if there is any doubt about the child’s survival, anyone can and should baptize the child.
If there is any question about the validity, it is good to have a conditional baptism in addition, properly administered by a Priest in Church.
I was baptized properly as a child in a High Anglican church belonging to an order of nuns. No doubt about it. My Aunt who witnessed it was still alive, and I had my baptismal certificate. But nevertheless I received a conditional baptism by a Priest when I became a Catholic—just to put it on record and be sure.
I was looking for this since I baptized one of my sons as he was turning blue in the oxygen tent. We did the follow-up as you suggest.
**If there is any question about the validity, it is good to have a conditional baptism in addition, properly administered by a Priest in Church.**