In John MacArthur’s sermon titled The Salvation of Babies Who Die, he explains his view of what happens when children die young.  You should listen to the message yourself including part 2, but here’s a summary of what he taught on the topic of salvation of children and the mentally handicapped including some answers to common questions on the topic

Summary
All children who die before they reach the condition (not age) of accountability by which they convincingly understand their sin and corruption and embrace the Gospel by faith are graciously saved eternally by God through the work of Jesus Christ being elect by sovereign choice, innocent of willful sin, rebellion and unbelief by which works they would be justly condemned to eternal punishment.

Background
1. Life begins at conception and these babies have eternal souls
2. Based on Psalm 139, God a) knows everything about me even before I can talk b) God is in complete control of my life c) God will never lose sight of me no matter what goes on. I can never be lost to God d) There is no circumstance that can in any way limit His knowledge e) The reason He knows me so well is He is my personal creator who has planned carefully my destiny.
3) Both Job and Solomon, in Job 3:1, 16-17 and Ecclesiastes 6 respectively, imply that babies who die young go immediately to a peaceful place (heaven)

At what age are children considered accountable?
There isn’t an age threshold, but rather a condition that matters.  Those who have not reached sufficient mature understanding in order to comprehend convincingly the issues of law and grace, sin and salvation are not accountable (Deuteronomy 1:39, Isaiah 7:16)

Are all children sinners?
Yes, all are sinful from conception (Psalm 51:5, 58:3, 143:2; Romans 5) or there wouldn’t be any basis for death

What implications does this truth of depravity have on the issue of dying infants?  Salvation is by grace just like it is for adults.  Adults have no more merit in their salvation than a child who God saves by grace.  It is God’s choice

What about infant baptism?
Infant baptism isn’t Biblical and has nothing to do with a child’s salvation

If infants are saved when they die, by what means are they saved?
They are saved by the sacrificial work of Jesus because that is the only means that anybody can be saved. God has predestined all He wills into salvation, including those in infancy. That salvation is by His sovereign choice through grace alone though all infants deserve eternal judgment because of their guilt and corruption. Their sins were paid for by the substitutionary sacrifice of Christ on the cross in which He bore the wrath of God not only for all who could believe, but for all who could not believe.  MacArthur points Charles Spurgeon and BB Warfield as supporters of this view

What does Scripture teach about damnation?
Scripture always, always connects eternal condemnation to the sinner’s deeds and the greatest sin is unbelief (Revelation 20:11-12, John 8:21, 24; 3:36).  There is no place in the Bible where judgment is based on any other grounds than the deeds of sin. It’s true they’re sinful by nature, but the account against them that condemns them is their deeds. God doesn’t charge people with actual sins until they commit them.

Conclusion
Children are born sinners. Their death proves that. But never being able to understand the truth and therefore consciously rejected and choose rebellion, they have no record against them in the books of God and they then constitute a marvelous and vast opportunity for sovereign grace to operate apart from any works at all. Salvation is by grace completely apart from works. Damnation is by works completely apart from grace. Infants have no sinful works to fill the books and condemn them. So if a baby dies, that baby is elect and instantly goes to heaven to eternal salvation and eternal glory.

If you have a little one that dies, rejoice. Count not your human loss, count your eternal gain. Count not that child as having lost, but having gained, having passed briefly through this life untouched by the wicked world only to enter into eternal glory and grace. The true sadness should be over those children of yours who live and reject the Gospel. Don’t sorrow over your children in heaven, sorrow over your children on earth that they should come to Christ. This is your great responsibility, your great opportunity.

For additional information on this topic see John MacArthur’s book Safe in the Arms of God.