How to improve chances of upgrade to paid economy Oneworld RTW ticket with Qantas FF points?

Looking at RTW paid economy ticket with Qantas and One World from Melbourne - London - Montreal - Washington - LA - Melbourne. My wife and I have 250,000 FF points and would like to upgrade to Premium or Business with Qantas on the Melbourne to London and LA to Melbourne legs. We are only Bronze status, so would we be subject to the priority system and put last in the queue. Is there any way we can improve our chances?

You could join Qantas Club or schedule flights on days where there’s less demand for the premium seats.