Well, the short answer from Scripture is anyone who dies without Christ is damned. But a more specific question is whether children, who can’t profess Christ (because they aren’t old enough), can nevertheless be saved BY him. We believe this is possible for all Old Testament saints. for example. And most Christians believe this can be true of young children and the mentally disabled as well.
Now, what’s important as we speculate on who may be saved apart from a personal confession of faith in Christ, is that we maintain the Bible’s GRACE alone, FAITH alone stance. We can’t go inventing new conditions of salvation. If we entertain a new condition for salvation (like if we say, kids are saved because they’re just so innocent, or my Buddhist aunt must have been saved because she was so nice) then we’re diminishing our sin and Christ’s necessary role as Savior, and making Jesus out to be a liar (John 14:6).
AND this simply puts a burden on people most will never reach. It’s by grace so NO ONE can boast and so that as many as possible can be saved. Who wants the condition of salvation to be by YOUTH alone? That’s bad news for those of us who are not children! Or by enough GOOD WORKS alone? That’s bad news for people, like me, who are so broken and sinful. So there are not many plans or conditions for salvation… there is only one: by GRACE alone, through FAITH in Christ alone. (Eph 2:8,9 & Acts 4:12).
Therefore, on what biblical basis (keeping the Bible’s grace alone faith alone stance in mind) can we imagine that young children will be saved who cannot authentically name Christ for themselves?
The case begins with the fact our Lord himself used children as an example of those who through TRUSTING HUMBLE, FAITH, are models of what Kingdom people are. He said,

“I tell you the truth, unless you change and become like little children, you will never enter the kingdom of heaven.”
Matt 18:3
This suggests that all little children enter the Kingdom.
What is it about children that becoming “like them” makes you eligible for salvation? We must conclude it’s based on their child-like faith (through the merits of Christ), not their own “youthful innocence” or their ignorance of the gospel. They are not innocent even if they were born without specific acts of sin – if we accept that “in sin my mother conceived me” (Ps 51).
Anyone who has raised children knows how early a rebellious streak shows up. Even toddlers can be taught a line, know it clearly, and transgress it very intentionally. In fact, Saint Augustine once said:
“If babies are innocent, it is not for lack of will to do harm, but for lack of strength!”
Confessions, Book I.7
So they aren’t innocent, even if they are ignorant of the gospel. So we mustn’t argue that children, (or the mentally ill or pagans in other nations without a Christian witness) aren’t subject to condemnation because they don’t have the Law or understand it or don’t clearly break it. The basis of our just condemnation is written on every human heart in our conscience regardless of whether we are too young and cannot read a bible, or born outside the reach of a bible.
Therefore, we must maintain that all are justly condemned and in need of grace (and this makes the preaching of the gospel the highest priority, in other lands and in our own households.) But having agreed to the need of even children to be saved, we agree with Jesus that they do seem to exhibit faith. And so they are saved by the merits of Christ based on the intrinsic trusting dependency of their youth and God’s prevenient grace.
In fact, rather than consider them cherubs of angelic purity, by Jesus calling them prototypical citizens of the kingdom, he was saying they were more like the “sick” he came for rather than the “well” whom he did not come for (Matt 9:12-13). Not surprising then that most children seem much more like the penitent tax collector than the moralistic, self-assured Pharisee (Luke 18): they are needy, rebellious, at times lost little sheep who fling themselves on the care of their Parents and find their trust rewarded with mercy.
Now, another biblical argument for why children are all automatically saved, is the “age of accountability”. While acknowledging the fact of our sin nature, we can see that this sin nature, while acted OUT from birth on, is never ACTUALIZED until an age when the child’s conscience becomes self aware of his personal participation in Adam’s fall. While there is no explicit argument in Scripture about what that age of accountability is, some feel the idea at least, is referenced by Isaiah in 7:16 (“But before the boy knows enough to reject the wrong and choose the right…”).
The age of 12 is most commonly associated with Jewish rites of passage into adulthood and “accountability”. Before this time (whenever it is) of being ACTUALIZED in his sin, the child is not accountable and therefore covered under what Augustine called “prevenient grace” – the grace God gives to all humans prior to the engagement of their will or choice to commit specific sin.
The above reasons might be taken together as a sort of cumulative case rather than separate arguments for the salvation of children. Regardless of how we approach this, knowing the heart of God is “not willing that any should perish” (2 Peter 3:9), we can more easily trust God with the unreached, or young children or anyone who cannot – through no fault of their own – make a personal confession of faith in Christ.