Skyflier: BA does use the Qantas Club (aka the Qantas Pub and you'll see why) in BKK. It's recently refurbished and down towards pier 5 - just turn right after passport control and keep going, the entrance way is huge and you can't miss it.
This accurate advice reminds me of one other thing for the OP.
It seems that BA and QF now largely operate from the aforementioned pier near the new QF Club. If transiting at BKK, you'll be unloaded onto the lower level of the terminal. There are lots of transit signs that are intended to make you walk all the way into the central part of the terminal before (a) showing you some shops in a poor "transit area" on the lower level and (b) taking you upstairs to the departure level, tipping you out into the shopping area near outbound passport control.
However, if you look closely as soon as you get to the terminal end of the pier and start walking along inside the terminal itself, there are some poorly marked doors to small staircases that lead upstairs a long time before you reach the central area. If your only destination is the QF Club, then use one of these and look for the Club when you get up to the departure level. It could save you a long walk to the shops and back if you're not interested in doing that.