In my restricted experience of 5 CX J flights last month, I was impressed with the CX Ground handling, especially out of HKG. Premium bags always came out first.
Furthermore, we flew LHR-HKG on BA and connected onto CX to CMB. Our bags were tagged as BA Club World, but when we arived in CMB, they had also acquired CX J Priority Tags during the HKG transit. So confirming what Guy said about CX handlers checking the class of travel on the manifest and tagging bags as appropriate.