Best F Avios Can Buy
 

I prefer JAL F.
CX F may be more convenient as it allows you to connect to other destinations easier.
Not bothered by Qantas or Qatar F.
I'd probably rank them in that order..

Sticking to "F [that] Avios has access to", I think the best is JL. JL F also has close-in availability surprisingly often on LHR-Tokyo though 3 seats could be a struggle; 2 is more achievable. JL Business is also quite fine.

CX is quite good, followed closely QF.

JL F service is very good. Extremely thoroughly attentive crews who practice the 'invisible elves" style of service - you're expected to use the call button at which point one will appear next to you within seconds. The ground service also does not suck, for example the gate agents will tend to find you in the queue and take you to the front. Clearly, They Do Know Who I Am :)

I don't rate QR and its robotic, terrified crew as highly as some people. I look under the veneer.

Regarding QR crew, I think the issue here is also there is also a massive gap between the way they typically act in J and the way they typically act in F and most posters are judging based on QR J. QR has a very involved service in J and on a full flight this makes it difficult for the crew to spend much time engaging in conversation with the passengers until after the meal service is complete, as the service is clearly quite labour intensive. In F, in comparison, I think there's much more time to get to know the crew and, frankly, I would rate the service I typically receive in QR F much more highly than what I used to get in BA F.

Regarding the OPs original question, I'm in the 'JL F is the best F Avios can buy' camp (particularly ex-TYO). CX is not bad, but frankly the catering is often no better than BA F these days (service, hard product and wine list still materially better). QR next, although I think it's basically impossible to book this on Avios now as QR is basically no longer releasing F class long haul redemption availability to partners. Then a big gap to BA, with AA bringing up the rear. I have never flown QF F, but I imagine it would slot in behind QR.

Don't discount QR F, I've flown it a number of times on the BKK route on an A380 and found it a very enjoyable experience. The hard product is miles ahead of BA F.

My preference would also be JAL, by quite a distance. CX F used to be almost as good, but has declined recently to the level where I'm not sure it would stand up to QR F on the 380...

Flew JAL F earlier this year LHR - HND, best flight overall I've ever taken, best sleep I've had on a plane and service was out of this World just resist the temptation to eat yourself stupid in the T3 OW lounges before you board.

I prefer BA F.... with a 2-4-1 ;)