Nelson was asked by the king to take a fleet into Naples, so I think he had his authority from those instructions. The Neapolitans were allies of Britain, and Neapolitan officers were placed under British command.
However, regarding the Neapolitan ship Minerva, this had first been under Troubridge, and then Captain Foote, before Nelson arrived and assumed command of the Allies in those waters. Foote, however, says that Thurn, the captain of the Minerva told him:
"I had written to the Count de Thurn for three galleys ... but the Count de Thurn at the same time informed me that his instructions were quite independent of my orders, and that he could not receive any but from his sovereign or those who were his superiors."
So it seems that the Neapolitans thought there was no "delegation of authority" there. Perhaps only when Nelson arrived did this change.
Presumably the king told Thurn to comply.
But, even if the Neapolitans were completely under British authority, Nelson could still have said "I can't convene a court-martial, Neapolitans are auxiliaries and I can't interfere".
What is interesting is that he did not choose to say this in June, but he had said it in January.