Q: I have a question about Baptism of Desire. What happens to the soul of the little ones who die without having been baptized. And can lay people baptize a baby through Baptism of Desire even years later after the baby has died. Like for instance a baby lost in a miscarriage?
A: Please check paragraphs 1257-1261 of the catechism. The desire for baptism brings about the fruits of baptism without being a sacrament (1258). A child whose parents were intending baptism surely fits in this category. 1261 does not say absolutely that the child is saved, but that we have a well-founded hope of their salvation.
As I read it, desire applies when it is manifest before the child dies, not years later. But hope always applies.