The F ticket (w/ Blue Exec Club status) will give you access to the Qantas First lounge in Singapore as well as the Concorde Room in LHR before your departure.

Whether you choose Qantas First on your layover vs Qatar's Premium Lounge is one for you to decide - my advice would be to prioritise a shower for comfort rather than food as you should be well looked after onboard anyway.

Edit: I should have added I think there's a small "The Bar" lounge by BA in Singapore as well which you'd get access to. I've got no insight as to whether it's worth a visit or not though (I googled and it looks like it's still open?)

I believe the inbound F boarding pass in SYD should grant you access to the Qantas First lounge in SYD which is a fantastic upgrade over the business lounge. If you're lucky ask for a spa/massage treatment - appointments are first come first served to non-QF F passengers and I've had a good track record of getting them with a bit of time in the lounge. In any case, you can get a nice breakfast and another shower in one of my favourite lounges.

Enjoy the trip!