My take:
HKG
I feel the best oneworld lounge in HKG is the CX F Wing, however with it under renovations that's not an option. I would go to the CX Lounge which is closest to the gate of your next flight (i.e. for the lower gate numbers or gates 500+, go to the J Wing; for mid-ranged gates, go to the Cabin; for high numbered gates, go to the F Pier).
If you need a shower, I can recommend the F Pier showers, which aren't bad. They do get quite steamy if you go for a hot shower since they are not as well ventilated as the F Wing cabanas (then again, if you're unfortunate at the F Wing to get a regular shower, then the F Pier showers win hands down since they are more spacious). The F Pier also has small personal relaxation rooms that have a recliner and a TV, so you can sit and watch some TV in peace, or use it to have a little nap.
The Cabin does not have showers.
The QF lounges are not bad but not as good as the CX ones. If they are not as popular as the CX ones then possibly they may be quieter.
If I had to take my pick, the QF F lounge is nice for atmosphere and the lounge environment itself, but the QF J lounge has larger shower rooms. Food is about par for the course on both (you can get a selection of two noodle dishes on demand in the QF F Lounge, or head a little further to the KA Gate 16 Lounge or the J Wing for a different noodle bar).
LHR T3
No real options here - BA lounge, which has a F and J (Club) side. As a QF WP you should be eligible for an Elemis spa appointment. Shower rooms are the same block for both lounges.
Entry to the special dining area in the BA F lounge is only for ticketed F passengers (on QF or BA), CCR holders, QF CL and BA Premiers.
I've read that the other two oneworld lounges - CX and AA - are both pretty average. If you want to lounge hop, go for it.
LHR T5
No access to the BA CCR unless you are flying BA-operated F services. QF codeshares on BA metal in F I think are OK, but QF F flights (even with BA codeshare) are not OK. Also, I believe you can gain access to the CCR if you have arrived into LHR on a BA-operated F ticket.
Otherwise, best lounge is T5 Galleries First South (main lounge complex). Can't really complain on this one. Showers located outside of the lounge near Elemis spa (in between Galleries First and the CCR).
LHR Arrivals
AA lounge in T3 only for ticketed QF F or QF J.
The QF website implies no access for QF elites or oneworld elites that are not on F or J tickets, however anecdotally it's been said that QF WPs have been able to get into the AA arrivals lounge.
Access to the BA T5 arrivals lounge is only for BA F or BA J pax.
It's important to note that arrivals lounges are not oneworld registered lounges, so all access arrangements outside of the carriers who operate the respective lounges are purely by agreement.
SIN
IMHO the QF/BA F Lounge is quite decent. It certainly doesn't have great food (except in the late evening when they start passing around little plates of food), they got rid of the water feature in the lounge and it can get crazy busy as all the QF flights start to get ready to go in the evening, but it's not all that bad.
Showers are decently sized, are located adjacent to the lounge entrance and are busy at the best of times. Be prepared to "get in line" at the shower reception and take a buzzer with you.
CX use the Skyview lounge, which is a step up from the QF/BA lounges, however I don't necessarily like it all that better because it is farther away from the gates used by QF/BA (not a problem if your onward is on CX), the food isn't leagues better (just more Chinese snacks, e.g. dim sum, and random rice dishes) and since it is an open lounge, you get the noise of the hubbub below the lounge. The Skyview lounge is 24 hours, so if you need to crash and don't feel like ponying up for the transit hotel or what not, you might be able to get away with it here.