The infant is sinful, but the sin is not imputed. Note Paul's statement: "I was alive without the law once."
So despite sin being present, the infant is spiritually alive and blessed with a real communion with God's presence or life on some level (being "alive without the law"). But the Law shows the growing child the reality of his or her sinfulness and so sin causes separation and spiritual death necessitating rebirth in Christ, walking in the kingdom of God.
I believe thus that the Christian child can grow up in the kingdom and so a precise "born-again moment" is unnecessary.
I think the special aspect of life we see in children "of such are the kingdom of God" is because despite their sin, it is not imputed and they are spiritually alive even in the Christian sense. In fact, I think people fail to recognize that some of the energy and life they feel growing up is the presence of God's Spirit.