And what if one of the spouses is strongly opposed?
The Church doesn't make the Sacrament invalid. It is the nature of the two parties involved for any number of reasons. The more popular one in the last 30 years in the US has been the 'not psychologically prepared to make this commitment', which I believe the Vatican might be slowing down acceptance of pretty soon. Others could be a previous marriage that was not admitted to, or anulled, a condition hidden from one of the parties that was later discovered; drug abuse, alcoholism, mental illness, where knowledge might have given the other party pause before making the commitment, inability or unwillingness to stay faithful to the vows.
As far as what happens if one of the parties doesn not want it annulled, I can't tell you. At that point, the divorce is final, so I don't know why the person would object to the annulment, unless they've been told, incorrectly, that it would affect the legitimacy of the kids. If the Sacrament is considered invalid because of the nature of one of the parties, it is invalid for BOTH parties, not just for the one about whom the judgement was made.