There is a provision in NJ law for the case where the inheritor is prohibited.
Firearms passing to heirs or legatees. Notwithstanding any other provision of this section concerning the transfer, receipt or acquisition of a firearm, a permit to purchase or a firearms purchaser identification card shall not be required for the passing of a firearm upon the death of an owner thereof to his heir or legatee, whether the same be by testamentary bequest or by the laws of intestacy. The person who shall so receive, or acquire said firearm shall, however, be subject to all other provisions of this chapter. If the heir or legatee of such firearm does not qualify to possess or carry it, he may retain ownership of the firearm for the purpose of sale for a period not exceeding 180 days, or for such further limited period as may be approved by the chief law enforcement officer of the municipality in which the heir or legatee resides or the superintendent, provided that such firearm is in the custody of the chief law enforcement officer of the municipality or the superintendent during such period.
So after firearms are found in your uncles basement, you or your aunts(??) SIL, can arrange to acquire them legally from the executor.