Since you are arriving on CX, I am not sure that you are entitled to use the BA arrivals lounge. Certainly, we weren't invited to do so when we arrived three weeks ago.

However, BA has opened a very nice domestic lounge where you can wait prior to your CPT flight (assuming you are taking the BA flight to CPT). They have a shower, 3 high speed internet terminals, a small array of food, and a lot of comfortable seats.