CathInfo.com

All valid Trinitarian baptisms are "Catholic baptisms".  Someone baptized by non-Catholics (remember, anyone can baptize validly, even an apostate) remains a Catholic until they attain the age of reason, at which time, if they embrace another religion (or no religion at all), they become at least material heretics, granted, through no fault of their own. 

It seems to me, then, that the Church creates a kind of legal fiction, by which those children are treated in Canon Law as never having been Catholics.  If it's not a legal fiction, then what is it?

"The term 'baptized in the Catholic Church' creates some difficulty, especially in cases of baptism administered by lay persons.  In the first place, if the father and mother, or at least one of them, are Catholics and adhere to the Church, the infant baptized at the request of the Catholic party by a non-Catholic doctor or nurse in a case of emergency may still be considered baptized in the Catholic Church, for there is but one baptism, and whether the reception of that baptism means the joining of the Catholic Church or of some non-Catholic denomination depends on the will of the person who has the right and duty to care for the welfare of the infant.  If neither parent adheres to the Catholic Church (i.e., if both are Protestants or apostate Catholics), but one of them consents to have the infant baptized by a Catholic priest, one must know whether some guarantee was given of the Catholic education of the child; if so, the child was by the will of the parent legitimately enrolled in the Catholic Church.  If such guarantee was not given, no Catholic priest or layman had the right to baptized the child, and it was not legitimately enrolled in the Church Church, except in urgent danger of death... The Committee for the Authentic Interpretation of the Code declared on April 29, 1940, that persons born of non-Catholics and baptized in the Catholic Church, but not raised as Catholics, are subject to the impediment of disparity of cult according to Canon 1070 when they marry unbaptized persons" (Woywod p. 713-14).

Mortara case probably deserves a thread of its own.  Very interesting.  That actually contributed to why Napoleon III turned on Pius IX, without whose support it became impossible for the Church to hold onto the papal states.