Use the CX lounge, in my experience it is never anywhere near as crowded as the BA/Qantas lounge. As a BA Gold member you should get the use of the First Class lounge. Since you are on CX you will be on the “lefthand side” of the airport. If you are leaving from gates 1-32 use The Wing – as soon as you leave the security/immigration area turn left along the wall adn there is a walkway straight into the lounge. If you are leaving from gates 33-80 then go to The Pier which is near gate 61 or 62 (the far end of the airport). The Wing has the cabanas (private bathrooms), The Pier has daybreak rooms which are private sitting rooms with their own TV, music on demand, and a natty little touchscreen from which you can call for more drinks. I haven’t tried The Cabin which is halfway along but that is combined business/first so you will effectively only get business class facilities.
If you are on the morning flight, you may like to know (little-known fact) that you can order a proper breakfast (eggs and bacon for example) from the waiters rather than just make do with what is on display