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.**
I baptized every one of my kids and some of my grandkids. They were then taken to the Church to have the paperwork completed where the priest chose to “make sure” I had done it right.
I had.
Matter of fact, it happened to me a second time. My mother came from a Florentine family. She was never religious, and never went to church, although my Aunt persuaded her to have me go to Episcopal Church as a child. Anyway, when she got old and my step-father died and she started getting senile dementia, I talked to her about the desirability of going to church, which would have provided her with what she needed at that time in her life.
She answered that she didn’t see how there could be a God, because there was so much pain and suffering in the world. I suggested that she read C.S. Lewis, “The Problem of Pain,” but I couldn’t convince her.
Later, she descended much further into dementia and had to go into a nursing home. I found a Catholic nursing home for her, partly because the nuns and nurses were so caring. She developed the habit of going to Mass in her wheel chair—maybe for the company, maybe for something to do. I asked her if she wanted to be a Catholic, and this time she said yes. Since I didn’t know if she had been baptized (non-religious Italians tended to go to church for baptism, marriage, and funerals, but not always), I talked it over with one of the nuns and a priest, and my mother agreed to be conditionally baptized. So for many years in the nursing home, she went to Mass, annual confession, regular communion, and last rites at the end. It was a wonderful ending to her rather painful life.