The first transfer AMEX -> CX tends to take a couple of days. Subsequent transfers seem to be fast, within a couple of hours.

With my first transfer, there was a redeem itinerary waiting for the AsiaMiles being credited to my account.

Based on the way the AsiaMiles desk reacted to my messages, I think, they could see the AsiaMiles already arrived (though not visible for me), so they only needed to wait out the period that the transfer might be canceled and from there go ahead.