The "they" references are confusing and matter a great deal.
I take it that because this is posted in the AA forum that this is an AA-issued ticket? Is it an award or a cash ticket? If it is an award ticket, is there availability on the CX segment (check the CX website)?
Either way the problem is with CX and AA cannot force another carrier to do anything. But, AA can ask a partner carrier to do so and has a OW liaison desk that interacts with other OW carriers, which all have similar desks for the same purpose.
Call back during regular business hours in the US. Start anew. When you get the front line agent's "no", politely ask for a supervisor and ask the supervisor to reach out to the OW liaison for you to request that CX force space open (presuming that there is space available and that this is award travel).
That will take several days, especially because this is a holiday weekend in the US. There is absolutely nothing you can do to speed this process up and you need the assistance of AA to make this happen.
While it may be frustrating, telling AA that it cannot do this will not help you and will simply lead to a flat out "no". So, be polite, but firm in your request.
Needless to say, if CX sold "your" seats out from under you, which it may have done to one of its elites, there won't be any space and that option is out.
While all of this is going on, I would suggest doing some research on AA/BA alternatives. AA can, of course, force space on AA metal and it can come close to doing so on BA. As the final option is always a cancellation for a refund and you apparently don't want that, it is well worth your effort to look to look at HKG-LHR for which there are oodles of options flying in both directions.