Will the French have to let the pirates go, according to NATO law?

I doubt it. Under international law, piracy is viewed as a universal crime, and every nation holds authority to exercise jurisdiction over the pirates, in spite of the flags carried by the ships involved and the nationalities of victims and pirates. The NATO mission working in the Gulf of Aden is specifically tasked with intercepting pirates. At some point, they will have to make arrangements to transfer them to shore to prosecute them, but they should not have to release them.