Infants go to heaven.
They are not accountable and therefore do not receive the imputation of sin and God is free to impute to them Christ's rightousness, not having denied Him. (2Sam.12:23, Rom.5:13,Jn.14:6.
That's not how the early Church saw it. Bishop Eusebius urged his parishoners to baptize as soon as posisble and not even wait for day 8 as the Jews did for circumcision.
The reason why Limbo infantum even became a hypothesis of the Church is because the Church was not certain that unbaptized infants go to heaven. The current Catholic Catechism takes the patristic view shared with the Orthodox Church's unbroken tradition, that we do not know, but that we trust God's mercy.
They are not accountable
No they are not, but unbaptized infants are not Christians.
What happens to unbpatized infants is beyond our knowledge or comprehension.