“As a Christian, do you think that adults from different religions are out of luck also?”
The question concerns infant baptism. What is your church’s position on unbaptized babies/infants who die?
The Catholic Church's position on this matter is that the issue cannot be defined as dogma one way or the other, since it is not part of the Deposit of Faith. Therefore, the Church doesn't have the ability to make a definitive statement. BTW: as an aside, at least this unwillingness to "create doctrine" without anything like a divine mandate should show you that the Church doesn't "just make things up out of wholecloth," as is often alleged!
Anyway, to fill in the obvious vacuum here, there have been several theories advanced over the years. One, quite rarely held these days, championed by St. Augustine, is simply that these infants are damned. Another, much more common today, is that the souls of infants can be given some kind of illumination and can make a choice, based on that, for God or no. Under these conditions, something like Baptism of Desire is achieved for those who accept God. The third option is probably the best-known and has had a very long pedigree. That is that infants who die without baptism go to a place of natural happiness, but are deprived of the Beatific Vision (actual presence before God in Heaven). This state is commonly known as "Limbo." It is sometimes considered to be the least punitive level of Hell, while others see it as separate. In any case, since the infants, and other small children, who haven't attained the age of reason at death haven't committed any personal sins, they are not punished with the physical pains of Hell as described by Jesus. To the extent that they're punished at all, it is through the knowledge of their eternal loss of the vision of God.
Those are the three main theories. There are some permutations of them, but you see the basic idea. Personally, I lean toward the Limbo theory myself, though it is not so much in favor, these days, and both JPII and Benedict seem to have opted away from it. But, since all of these theories are just opinions, anyway, a Catholic is free to hold to any one of them in good conscience. As the Catechism of the Catholic Church says (1261), all we can do in this scenario is "entrust them to the mercy of God, as she [the Church] does in her funeral rites for them."