I found arrivals and passport control in disarray when I landed on this trip - a 40-minute rugby-scrum wait behind a non-EU flight bearing migrants. There was a glitch checking in for departure too, with IT problems at the check-in desks. Otherwise departure went quite smoothly, though the use of buses and the time spent hanging around on buses, is always annoying.
If you are flying BA Club Europe or are a BA Executive Club member with silver/gold status (I'm not sure about bronze), be aware that there's a convenient 'Fast Track' priority route through security in departures. This is marked off to the left as you approach the security screening area. The fast track option wasn't mentioned at check-in, so I'm not certain who exactly is entitled to use it, but I was waved through with my Club Europe boarding pass. You can also pay for access (€5). Electronic signs indicated a 10 minute wait at security, but the Fast Track was empty and I sailed through in moments.
The lounge for BA customers at Naples airport is shared with other airlines, and passengers can also pay to enter (€22). Located by Gate 17, the lounge is still the wrong side of final passport control, so allow time to get to your gate. It's very small, with armchairs, reclining loungers but no workstations and no dedicated charging points for electronic devices, although there are a few randomly-placed sockets if you hunt around. It was crowded when I arrived, to the extent of strangers having to share sofas. At the back of lounge are a couple of toilets and a shower.