The priest should still apply for a decree of nullity for a Catholic who was married civilly. Its a very simple process where the evidence of the civil marriage and divorce are forwarded to the tribunal and it recognizes that its an invalid marriage. It can all be taken care of within a week. There is no detailed investigation or witness statements like in a full annulment case where someone was married in the Church.
However, even if the priest failed to get the decree of nullity for the civil marriage the sacramental marriage would still be valid. The reason is because the civil marriage was invalid and therefore no marriage bond exists to prevent the Church marriage from being valid. Decrees of nullity do not invalidate a once valid marriage, they simply recognize that no valid marriage bond existed.