OK, so I've found out a bit more about this. For the time being, BA's lounge is the Blue Lounge (where I am now) and that is run by Groundforce, the handling agents for BA.

A new ANA lounge is being built "in the summer", which BA will use, but the work has not yet started. According to the oneworld site, this lounge is open and landside, but they have jumped the gun there. I don't think it will be landside either, just a hunch on my part. Manage My Booking correctly show the Blue Lounge as being the BA lounge for today's flight. US / AA and PP can use the Lisbon Lounge, which is nearer the central duty free area, whereas the Blue Lounge is on the left side (facing the apron) towards gate 12. ANA in this context being the airport operators, not the airline.

There is an ANA lounge landside, on the lower level by the car rental desks, but I can't believe that's being primed for this (it also entirely enclosed with no windows, it's currently used as a pre-security upmarket holding pen for VIPs.

Anyway it is the Blue Lounge for now.