There is absolutely no chance whatsoever that any unbaptized child who dies will ever enter into the kingdom of heaven. At all. Ever.
It was the heresy of the Pelagians who first proposed that unbaptized infants who die could have blessedness, but not eternal life.
For over 800 years after Augustine, the Church endorsed and taught, in its ordinary and universal magisterium that Infants who die go to hell where they are tormented not only through the lack of the vision of God, but by the pain of sense. But their torment is the lightest.
Now, before everyone here loses it on me, please consider something:
The scholastics got a lot of things right. However, their methodology presupposes the acceptance of certain philosophical tenets that are not necessarily compatible with divine revelation. We are to accept revelation FIRST, and, ideally, philosophy is supposed to help explicate it.
Now, in regards to this issue there is a fatal flaw I believe people make regarding what is experienced after death. THat is, they often fail to take into account the general judgement as well.
Remember, our souls will be reunited to our bodies, and we will be judged before all. In this final judgement, there are only two alternatives, Heaven, and Hell.
Divine Revelation tells us in the book of Revelation what will happen:
Revelation 20:11-15
11And I saw a great white throne, and one sitting upon it, from whose face the earth and heaven fled away, and there was no place found for them.
12And I saw the dead, great and small, standing in the presence of the throne, and the books were opened; and another book was opened, which is the book of life; and the dead were judged by those things which were written in the books, according to their works.
13And the sea gave up the dead that were in it, and death and hell gave up their dead that were in them; and they were judged every one according to their works.
14And hell and death were cast into the pool of fire. This is the second death.
15And whosoever was not found written in the book of life, was cast into the pool of fire.
So here it is clear that the great and small are subject to the SAME judgement, and the same KIND of torment.
Now, if a person is not baptized, his name is not in the book of life. Therefore, an unbaptized infant, who is a sinner from conception, is NOT in the book of life.
Now, if in the END they share in the eternal suffering of fire with the rest of the damned, why should we believe they suffer less IMMEDIATELY after death in the particular judgement?
It would not be just for God to mitigate their punishment with natural happiness only to surprise them with fire later. They could not do anything to merit FURTHER punishment, so why would it increase in the end?
But since we KNOW the fate of the unbaptized, we must conclude, in the name of justice that those who merit one kind of punishment in the end (the lake of fire) can merit no less in the particular judgement.
Therefore, immediately after their death, unbaptized infants must descend to the fire of hell.