The Church teaches that it is unlawful to receive baptism from anyone but a Catholic priest, but this is referring to Catholic parents looking to have their children baptized. If one accepts the Catholic faith, and therefore the ecclesial laws that state receiving baptism from non-priests is unlawful, then it is just most proper for them to receive this sacrament (which opens the door to the other sacraments, by bringing a person into the Church) from a priest.
However, the Church also teaches that anyone has the power to baptize - even unbaptized persons and unbelievers. And so in an emergency situation, where no priest is around, a cooperative nurse or neighbour could offer the sacrament. For it to be effective, they must simply use the correct formula: "[Name], I baptize you in the name (singular) of the Father, and the Son, and the Holy Spirit" (the form of the sacrament), and pour cleaning water over the person's head (or submerge them) while doing so (the matter of the sacrament).
So it is not the case that only the Apostles and their successors have the ability to baptize, according to the Church. Consecrating a host into the Eucharist, or offering Confession, is a different story.
Okay, well this is interesting and not at all what I was expecting.
It is thus that the Church teaches that baptisms that Protestants receive - so long as the matter and form of the sacrament are correct - bring them into the Catholic Church. "One Lord, one faith, one baptism" (). Either the baptism accomplishes nothing, by virtue of the matter or form being incorrect, or it brings one into the Catholic Church - albeit an imperfect communion, since upon reaching the age of reason the baptized will not necessarily assent to all those "range of facts" which constitute the Catholic faith.
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