We’ll assume both were free to marry, and are now civilly divorced.
On its face, you describe what is commonly called a lack of form marriage. The Catholic was bound to wed in the presence of an authorized priest or deacon (rarely, a lay person can be delegated under certain circumstances). If he or she did not, it would not matter whether the non Catholic person was baptized or not.
Generally, with three exceptions I note below, a simple process that is not an actual annulment process must be followed.
There is an investigation and declaration by a competent Church authority of some kind. Most bishops assign the tribunal or another curia office do this. It involves gathering documents of baptism, marriage and divorce. (In a few places, the bishop just lets the normal pre-marital investigation settle the issue. According to the Pontifical Council for the Authentic Interpretation of Legislative Texts, the pre-marital investigation itself suffices.) Your parish priest will know the practice of your diocese.
Here are the exceptions. If one is present, an annulment process is needed:
-they had received a dispensation to do this from the diocesan bishop
- the Catholic had formally defected from the Catholic Church (this is complex, and the tribunal or curia will need to figure this out)
- the Catholic married an Orthodox in the presence of an Orthodox priest using sacred rite.
Certainly if the couple wishes to convalidate their marriage, no declaration of its nullity needs to be done first.
A couple who married using the extraordinary form of marriage (marooned on desert island, no authorized cleric, etc.) would require an annulment, since the form, while extraordinary, is recognized by law.
(I’m a defender of the bond with the tribunal of the diocese mentioned below.)
God bless,