Stop.

This entire problem is caused by OP not waiting for CX to follow through with BA as it offered to do.

The ticket now has an "illegal" connection at LHR and is invalid. It is caused by BA, but as CX is the ticketing carrier, it is for CX to fix the problem. In order to do so, CX must contact BA. This is a manual process accomplished throw the OW liaison operation at CX and then BA. It may take several days. There is no guarantee that it will suceed.

But, OP did not wait. He started making other calls and that has now caused him problems because he was "disappointed" that he would have to wait. As the travel is not until next May, he should simply calm down and wait.

OP - Call CX back and see if you can get CX to start the process all over again. Ask CX to contact BA on your behalf and ask for space on the flight you want and for CX to call you back when it hears back. If CX is able to get authority, you will have new tickets and all will be well. If it is not, you will have to take one of the other options.

Do not start calling other places while this process is pending. That is how ticket problems are created.