If the parents were non-Muslims at the time of conceiving the child, and if the relationship between them was legitimately recognized by their religion, then the child is considered to be from a legitimate birth and can lead the prayers, if the relationship was not religiously recognized in their religion, then he cannot lead the prayers even after embracing Islam. If the parents were Muslims at the time of conceiving the child and the relationship was governed by a religiously-accepted marriage contract - even if it is verbal and not recognized by the state - then the child is recognized as their legitimate child and therefore he can lead the prayers, if there was no such religious contract, then the child is not from a legitimate birth and then cannot lead the prayers.