I'm not a BA flyer, but this thread came up when I searched Turin lounges, so I thought I'd update it. Evidently, TRN only has one lounge airside, the one I am in as I post. It's called the "Welcome Lounge." It seems to be only for passengers on AZ, BA, and LH. I asked the agent at the desk if alliance members were allowed in. She no, only passengers on the above airlines and that they must get pass at checkin.
This one is nothing special, in fact it makes me feel better about the state of lounges in the US. Very little food: stale sandwiches and croissants, bags of chips, soft drinks, beer, bad white wine, spumanti, and a few bottles of cheap liquor. The best offering seems to be Courvoisier VS. Seating is very limited, but It's not crowded on this Saturday afternoon. There are four chairs facing the tarmac, three sofas with their backs to the window, and a few other randomly placed chairs. As has been mentioned, the terminal is such a nice one that I would probably forego this lounge except for the wifi, for which I had to pay .50 E.
Restrooms are outside of the lounge.
There is also an AZ Salon VIP which is small, but quite nice--not much more in the food and drink departments than the one I am in, but it's very stylishly decorated. It also has restrooms in the lounge. All of the SkyTeam airlines' logos are on the door, but I don't know what credentials you need to get in. It probably depends a lot on who is at the desk. I have been in as an AF J passenger three times, and each time the attendant has looked at my boarding pass, heaved a heavy sigh while rolling eyes, and waved me in. Its biggest drawback is that it is landside. There is a "priority" security check next to it seemingly only for the lounge passengers, but the last time I was there, several families who were not experienced travelers were in the line and I almost missed my flight.
There is a lounge which Priority Pass uses landside near the checkin desks. I haven't seen it, but have heard it's pretty dreary.