The fare differences are probably from different ways they price things - they do not have a cartel

I find NF (which you are guaranteed on BA) is much nicer and more private than JL, although I find the food to be better on JL (OK, I admit, it's much better on JL - if only BA could improve the food to that level!). Communication would be easier on BA unless you happen to speak Japanese, to state the obvious. This may actually be a bigger factor in F in terms of service refinement etc than you might think. The style of service is very different too - you might find JL attentive but possibly perceived to be a bit distant, partially due to the language issues but more because they have a more 'rigid' customer-facing rules I believe, whereas on BA you can joke and carry on quite a bit if you wish to do so (it's not that you can't joke with JL crew though).

You'll have access to CCR on BA but not on JL ex-LHR but you'll have access to the identical lounge on the way back. You could even call a travel agent and book BA ex-LHR and JL ex-TYO for instance, to try both out.

I haven't tried ANA F so I can't comment on it but I generally find their premium class food to be better than JL's, although I can't comment on F or the LHR-TYO route. But the hard product on BA is very nice and it all depends on what your priorities are.