The issue is, if both he and his first wife were unbaptized, then their marriage was civil/natural only.
If he becomes baptized and joins the Church, he could marry *again* (technically it wouldn't be a second marriage, but be his first SACRAMENTAL marriage) to a practicing catholic. This assumes the original spouse does not want to join the Faith.
A civil/natural marriage can be "upgraded" to a sacrament in this specific circuмstance, as St Paul explains. Since the purpose of marriage is to get to heaven, a civil/natural marriage in which a spouse is hostile to the Faith can be "dissolved" (not sure if that's the correct word) so that a sacramental union can take place.
Would this need the approval of a marriage tribunal? A catholic marriage tribunal wouldn't waste their time on non-sacramental marriages, as it's clear in Canon Law that these aren't catholic marriages. I've heard these cases handled by Trad priests many times. A parish priest may have the power to decide, but I'm unsure.
If I remember correctly, it can be handled at the diocesan level, however, again, one cannot just decide this for himself, UNLESS we can find that it is explicitly allowed to do so. So, a Canon Law commentary on these canons should clearly state that a layman can, when lacking a priest or bishop with ordinary jurisdiction or a marriage tribunal, presume his non-sacramental marriage dissolved.
I'd be extremely surprised if that was the case. Why? Because the marriage has to be
dissolved, that's an act that has to take place, so, it's not that the marriage automatically dissolves once someone converts and would like the issue to go away, but it has to be dissolved by a competent authority, which is not present.
In any case, StrivingCatholic should first come to terms with the Great Apostasy, the question of the Pope, the New Mass, etc. otherwise we're putting the cart before the horse.