Perhaps the "Board of officers" and the frigate "temporarily neapolitan" were both induced by the situation:
- an HMS could not host a "Court Martial" of another nation, this would have been uncompatible with sovereignity of the British Navy.
- but Nelson had not much other options at hand:
Neapolitan rebels, protected in Naples castles by french garrisons, had surrendered to Cardinal Ruffo, under the condition of being transferred to polacca (local transport ships), free to sail to french-occupied Italy. Lord Hamilton and Nelson arrived with an english squadron and decided not to comply to this surrender, traping rebels and their families in the harbour.
So transferring back prisoners to Ruffo, representative of the Napolitan king who was bound by his word as an aristocrat, a stateman and a chuchman, was not an option. Only condamnation by neapolitan officers under british patronage was feasible.
Arriving the next day, king Ferdinand confirmed Nelson's decision, sending Ruffo in disgrace, and prisoners to gaols.
Why Nelson wanted Carracciolo to be hanged before the King arrival is a mistery; perhaps to make sure that the feeble-minded king would not act with leniency, as Ruffo had started to do, due to the high social status of many republicans leaders.
Alexandre Dumas, whose father was a collateral victim of Neapolitan civil war, wrote a strongly biased, but extremely well documented novell based on this event, perhaps his major book : "La San Felice"