So, I just came back from LIN, and it does indeed seem that the renaming/refurbishing didn't make a difference: the lady at the Alitalia lounge reception desk told me that Oneworld Emerald and Sapphire flying on Air Berlin have access.
In theory, anyway. Upon showing my BAEC Silver card, she told me that I needed Gold or Platinum. She was obviously referring to Topbonus, and even though she agreed that Oneworld Sapphire should get me in, she didn't accept my BA card. She then asked a colleague on the phone who agreed that I needed Gold or Platinum.
I didn't want to make more of a fuss than I already had, especially as there was a perfectly adequate BA lounge downstairs, so I left it at that. (She kindly told me that I should come back in case the BA lounge didn't let me in either and then she'd sort it out with a supervisor.)
The Alitalia lounge did look a bit nicer (it has windows and almost certainly better coffee) and I'd have liked to try it, but I can't really blame their staff for not knowing the intricacies of Oneworld's frequent flyer tiers.
I'll see how it goes in FCO in a few weeks where the BA lounge isn't quite as easily accessible for Schengen flights.