That's one of those things the Church does not have a dogmatic answer for. On the one hand, Christ taught that baptism is necessary for salvation.
On the other hand, people who weren't baptized (e.g. the "Good Thief")(and for that matter, Moses, Elijah, etc.) were clearly saved, which prompted the Church to teach about "baptism of desire" (meaning, if you desire to do all that God requires of you, but you die, God is just and accepts the implicit desire of your heart.)
And on the other other hand (I have three hands here? OK) babies can't desire anything, but Christ did say that we had to become like little children to enter the Kingdom of Heaven. He does seem to have a strong liking for children. (!!)
The Catechism says this:
"1250 Born with a fallen human nature and tainted by original sin, children also have need of the new birth in Baptism to be freed from the power of darkness and brought into the realm of the freedom of the children of God, to which all men are called. The sheer gratuitousness of the grace of salvation is particularly manifest in infant Baptism. The Church and the parents would deny a child the priceless grace of becoming a child of God were they not to confer Baptism shortly after birth."
The Catechism also says this:
"1261 As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them. Indeed, the great mercy of God who desires that all men should be saved, and Jesus' tenderness toward children which caused him to say: "Let the children come to me, do not hinder them,"64 allow us to hope that there is a way of salvation for children who have died without Baptism. All the more urgent is the Church's call not to prevent little children coming to Christ through the gift of holy Baptism."
Some have speculated that maybe God gives babies a moment of infused knowledge in which they can choose for Him or against Him. Other speculate that they are given an existence of perfect natural happiness at the "threshold" of heaven, so to speak threshold = limina in Latin, from which is derived the term "limbo") --- but we don't know.
Frankly, it's mysterious. We don't know exactly how God's mercy works.
Thanks for your reply, my 3-handed FRiend.
Some Catholics have held that babies who die without baptism do not go to Heaven, but do not go to the torturous Hell of the wicked, either, but Rather, that there is a Hell that is more similar to the Hell once known as Abraham's Bosom, where the souls remain with neither torment, nor the completion of the beatific vision. But I'd like to believe that the Church shall storm this Hell, and bring all those in it to a beatific vision. "And the Gates of Hell shall not withstand it."