I’m not surprised your friends are getting conflicting opinions on this question, because even many pastors, let alone other parish staffers, don’t understand Church law on this point. The confusion arises because the rules in these cases seem to differ for parties who are Catholic from those which apply to parties who are not.
In brief, if the facts are as you describe them, your friends will not need to have their marriage to each other convalidated by Church authorities upon becoming Catholic. As soon as both annulment petitions are formally granted (assuming this is done and assuming there were no other unusual factors present—your letter suggests none), your friends’ marriage to each other will automatically be recognized by the Church and will be presumed to be both valid and sacramental.
Here’s why: At the time your friends married each other, the only canonical obstacle to their wedding was ligamen, the fact of their prior marriage bonds (canon 1085). But if annulments are declared for both prior marriages, that means that, at the time of their marriage to each other, your friends were canonically free to contract marriage, and the manner in which they chose to marry would have been lawful for them at the time. Thus, their second marriage could be recognized without any further qualifications.
How would this question differ for Catholics, thus giving rise to the confusion? Well, if two Catholics, previously married to others and subsequently divorced, sought to marry in the Church, virtually any priest would have told them “No way, not without an annulment” (canon 1085 again). Unfortunately, such Catholics not infrequently then turn to civil magistrates for their wedding. That kind of wedding ceremony would be a violation of the requirement of canonical form which binds most Catholics (canons 1108 and 1117), meaning that, without an annulment followed by “convalidation” (canon 1160), such a second marriage would not be recognized in the Church.