An annulment does not "break" a marriage, as does a divorce. It declares that the marriage never happened in the first place.
This meant that the married couple would revert to their previous legal status (the "wife" would revert to being a "spinster" if she hadn't been previously married).
However, any children from an annulled marriage would still be considered legitimate in every sense, unless this had been declared otherwise in law. As Harold J. Berman observed:
Where the parties married in good faith, without knowledge of an
impediment, the canonists held that the children of the marriage were
legitimate and that the marriage itself was valid up to the day it was
Interestingly, in most cases, children would remain in the custody of the father. We have examples of this in the case of the daughters of King Louis VII of France and his wife Duchess Eleanor of Aquitaine. Louis and Eleanor were eventually granted an annulment of the ground of consanguinity, and their daughters Alix of France and Marie of France were given into the custody of their father. In this particular case, the girls were declared illegitimate in the courts, but not because of the annulment.
The question of the legitimacy of the daughters of Henry VIII was entirely to do with the Royal succession. Of course, there were some - particularly in continental Europe - who did not recognise the legitimacy of the marriage of Elizabeth's mother, Anne Boleyn, to Henry VIII. For these people, Elizabeth had never been a legitimate daughter of Henry in the first place.