BA does a very credible job on both JFK-LHR and LHR-DXB, CX is generally excellent, and RJ gets mixed reviews, so I would suggest BA and CX.

The BA JFK lounge is very good, especially considering that it's not a home airport, and the LHR transfer mess is somewhat mitigated by the fact that your transfer is T4>T4.

Cathay's C class should be significantly better than RJ's offering. Their lounges in HKG are superior.

See http://www.flyertalk.com/forum/showthread.php?p=8090242 and the link inside the thread for further opinions and reports.