Don't know about CX/AM.

I had a bad experience with using BA miles (before they were called Avios) to book CX.

We booked two CX F tickets YYZ-HKG. CX downgraded the aircraft and F was no longer available on our flight. We wanted to rebook on a different CX flight where there were 2 F seats. But BA could only see one although AM website showed there were 2.

That's fine. I asked BA to split the booking and we could travel on two different dates. BA refused to split the booking and rebook us free of charge. Our only option was to cancel. And we then rebooked with 2 separate bookings.

Finally, when we looked at the refund (of the YQ, Taxes etc), they deducted a refund charge from the total. It was fine after I disputed it with Amex. But what a crook this airline BA is!