I’m sorry for the nervous breakdown that your family member experienced. I have addressed this question several times over the years, but it remains an unresolved issue for many Catholics. How else can we explain why the same question is asked so frequently?
People are saved by the passion, death and resurrection of Jesus Christ—even if they did not have an explicit faith in him.
The Catechism of the Catholic Church teaches: “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 [people] should be saved, and Jesus’ tenderness toward children which caused him to say: ‘Let the children come to me, do not hinder them,’ 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” (#1261).
The Church’s teaching about limbo (a place of natural happiness) was never a defined article of faith, such as the full divinity and humanity of Jesus or his presence in the Eucharist. Limbo was a common theological opinion that tried to reaffirm Jesus’ role in the salvation of people while stressing the importance of the Sacrament of Baptism. Both of those elements are still true even if the Church does not ask people to believe in limbo.
For whatever reason, some people rather easily came to the conclusion that a just and loving God would not rule out eternal happiness for a child who died before being baptized. Informally, they long ago came to the same conclusion that the Church officially teaches today.
Was your family member’s worrying all for naught? That family member and the child whose death prompted the worrying were always in God’s loving care. Only God knows a person’s heart completely and thus only God can pass judgment on each of us.