Obviously, an anullment would need to find that the first marriage of the Catholic was invalid before a marriage could take place.
As for the second marriage, it will probably be investigated either way…but the process will be different based on some things.
Was the non-catholic and/or their previous spouse baptized or not?
If both of them were, it could still have been a valid sacramental marriage (non-Catholic christians are not bound to the Catholic marriage rite)…if it was not consumatted, the Petrine Priveledge may invoked to dissolve it if the argument can be made that it should be dissolved. This could be pretty hard as it involves proving it is not consumated and petitioning the Pope.
If one was baptized and the other wasn’t, it could still be considered a valid natural marriage…though probably not sacramental. Theoretically, only a valid, consumated, sacramental marriage is entirely indissolvable by anything but death. Under natural law, ending a purely natural marriage is hypothetically possible; the Mosaic law allowed divorce of its purely legal (non-sacramental) marriages, after all. But the Church usually does not (in fact, I think never) recognize the dissolution of even purely natural marriages because of Jesus’s strong words against any divorce, and so for a person non-sacramentally, but still validly, married before, the Church still seeks to establish that the first marriage was invalid before allowing them to remarry a Christian (as for a non-Christian previously married to a non-Christian marrying another non-Christian, the Church doesnt get involved…ending purely natural marriage is theoretically possible so the Church doesnt speak on the issue if all the parties are non-baptized, but in allowing baptized people to marry the Church never assumes any valid marriage, sacramental or not, has been ended unless the spouse dies or it declares the first marriage invalid), however, though again since it was not consumated, the Petrine priviledge could be invoked…and probably much easier than in the first case because it was only a natural marriage and the person wants to marry baptized Catholic.
If niether was baptized it should be pretty easy. Again, while ending a purely natural valid marriage is hypothetically possible, if someone previously in a valid though non-sacramental marriage wants to be married to a Christian…the Church requires invalidity of the first marriage to be established just to be safe and to encourage even purely natural marriages to stay together, as was the Creators original design. But establishing that purely civil marriage that wasnt consumatted was invalid should not be hard. And even in the rare case it is found valid (although not sacramental), because it was not consumatted the Petrine priviledge could be invoked. And while this involves petitioning the Vatican…because it was two non-baptized people in a civil ceremony…it should be relatively easy to obtain.
But there is an easier way if niether was baptized at the time. Under the Pauline Priviledge, if one of the parties gets baptized, then they may remarry a new baptized partner because a sacramental valid marriage supersedes a merely natural valid marriage…and their natural old marriage is dissolved by the very act of the new sacramental marriage taking place. But under canon law they must ask if the old partner wants to get baptized and have their marriage sacramentalized and blessed first. But in this case, since that first partner already agreed to get a civil annulment, that shouldnt be a problem.
