You are right. The check in counter agent could not tell me the name of the lounge, and I mistakenly relied on LoungeBuddy, which indicated the AA lounge. I just checked both the BA website and LoungeReview (a better if not user-friendly solution), and they shoed BA had access to Extime. This time I would not have had much time to spend there anyway, but good to know for next time.

And I've deleted my LoungeBuddy app. They just don't support it anymore.