We are ALL born with a sinful nature. Babies are innocent but they ARE sinful because of the nature with which they are born into this sinful world. I doubt very much that there are ANY babies or children in hell, not even in limbo.
The Catohlic Church says that they are innocent of personal sin but carry the Original Sin. The Orthodox would also say that the children are innocent of the personal sin, but they would qualify the condition they inherit from Adam differently and not call it sin.