Map of Changi Airport! As above T1, T2 and T3 are all connected. You can walk or take the air train to get between them.
Security at Changi in T1, T2 and T3 is all done at the gate. The barrier between groundside and airside is immigration.
I have not been to the swimming pool in T1.
In terms of which lounge to go to I don't think CX will be possible any more as they have moved to T4. Between BA and QF it's very much personal preference. The good news is they are right next to each other. If you want a proper feed then QF is better as they have proper dining in there, if you like a can of Guinness and other self serve alcohol as well as usually being a bit quieter then the BA lounge is the best bet.
