MAA update: the new Terminal 2 has a TFS lounge (described as "temporary" on the Oneworld website). After security and immigration. Standard BA/OW rules.
They use a paper invitation system, handed out at check-in. But when I (mistakenly) thought I'd lost mine at security, the agent was happy to take a photo of my boarding pass.

The lounge is quite decent. Fairly busy for 0400, but there were plenty of seats available. It may well be overwhelmed at other times, but if you're there for BA36 you should be fine. Hot buffet, staffed bar, staffed coffee machines (not espresso, but they'll push the buttons for you).

The lounge has toilets, though they aren't obvious: out the door by the bar marked "Fire Exit", and down a corridor in a currently less-finished part of the building. But large and clean.