Tickets and reservations are two separate things (although sometimes linked). It is perfectly possible to make a reservation without a ticket or to have a flexible ticket coupon without a reservation. If you have a full fare ticket for LHR-HKG on, say, BA ticket stock and you wish to travel on CX (and, frankly, who wouldn't especially now that BA is shafting everyone with new baggage rules?) then all you need to do is call CX to get a reservation on the flight you require in Y, J or F (whichever full fare class your ticket is in) and give them the BA ticket number.

In principle, any airline should accept a full fare ticket regardless of which airline issued it. In practice, because there are so many different sets of ticket rules these days, and it isn't always clear what is an absolutely open full-fare ticket, the carrier you wish to travel on may request that the carrier which issued the ticket endorse it for use on different carriers, in which case you would need to get BA to put a sticker on the ticket coupon (if paper) or endorse it electronically (if e-tkt). Of course, BA being BA they might try to charge you GBP25 for doing this... and in principle it isn't really necessary since the official IATA ticketing rules are (I believe) that a ticket is valid on any airline that flies that route unless there are endorsements that limit that validity.