Even if the Church granted the annulment for improper or nebulous reasons, as long as the petitioners for annulments did not lie and obtain it under false pretenses, the annulments would be valid. No matter how sinful a priest may be, he doesn't lose his priestly abilities unless he is officially laicized. The fault is with those who too freely grant annulments, but the annulments remain valid.This is not to say that many don't lie. For them, the annulment would be invalid, and they will have to answer for it.
I believe the ruling would still be invalid, but they would not be guilty of adultery of they re-married others:
What if on a later day they read up on annulments and realize they have a bogus ruling. Then I would say their innocence is gone. If they were re-married, they would have a problem, because they would be have the awareness of living in adultery from that day of enlightenment forward.