I'm confused. JFK-LHR-HKG BA F will give you access to both Concorde Rooms, but HKG-LHR-JFK will only give you the LHR Concorde Room. But your OP is about a different route?

If CX direct in F, no Concorde Room access at JFK.

Don't swap CX for BA just for the Concorde Room - BA catering out of HKG is not great, and CX is better in all regards.