Paladins Code, Paladins Moral Code, Paladins playing Lawful Good.
These discussions pop up ALL the time, in virtually every single gaming group I've had the pleasure or honor of playing with or observing.
Quite simply, YOU as the GM need to sit down with the player and make absolutely sure that you and he are on the same wavelength about the nature of the reasonings behind a paladins actions in your game.
Quite simply put: The Paladin was both RIGHT and WRONG, depending solely on how YOU define Lawful Good and the Code of the Paladin in YOUR game.
If you define a Paladin as a servant of the church, invested with divine powers, but still under the control of the church - then this Paladin is in clear violation - he's willfully disobeyed the orders of a superior in the church (assuming the cardinal would be a superior)
If, in your game, a Paladin is a direct servant of the gods (i.e. an epic paladin is second only to a deities avatar), then this paladin is clearly well within his rights to ignore the cardinal and lay waste to the evils of the false children.
In simpler terms - the Paladin did exactly right in Good vs Evil - the false children were not children, they were evil beings in the guise of children, regardless of how physically harmless they were, they had done heinous things and had attempted to control his mind. So yes - in Good vs Evil, the Paladin is clearly within his right. [edit: assuming of course, he had exhausted the possibility of converting them to good - better to weaken evil and strengthen good than just to weaken evil]
Where it gets sticky is the Lawful vs Chaotic. Was it lawful for him to destroy the children? Was it lawful for him to ignore the request of the cardinal? Was it lawful for him to ignore the request of his fellow party members? (If the wizard had some form of authority over the paladin, then absolutely not on the second kill)
As I said, it depends largely on your intrepretation of a Paladin in your game. The PHB is deliberately vague and open to interpretation. Go find 2 people who do not belong to the same church - they will, by nature, disagree on things like morality and divine law.
That's why you need to sit down with your Paladin player and hammer out what your world expects of Paladins. Some game worlds expect Paladins like Sturm of Krynn, others expect Paladins of a different sort, ones who answer to the church not themselves or a higher power.
This is easily worked out with a player - just sit down and figure out who he thinks he answers to and correct him if necessary. Compromise would undoubtedly be good here, btw.