E-ticket receipts are issued in (roughly) order of priority, so if your flight is not for a long time into the future you may not receive it for some days, but if your flight is tomorrow generally you will get it almost instantaneously.

My suggestion is to call the Platinum desk, but ask them to call the ticketing desk to get the ticket issued while you are on the phone. You could say you need it now because of an upgrade request, or you need it for work or somesuch reason to say you need it now. Some parts of QF have normal business working day operating hours, I am not certain this will work on a weekend. But it can be done.