I have been travelling to and from Japan for many years and from London it is very difficult to get any reward availability on most of the carriers you mentioned and there is no easy fix apart from determination, persistence and a lot of flexibility in when you travel!

The OneWorld label doesn't necessarily equate to a joined up or customer focused experience for redemption bookings but you could get creative by looking at Seoul as an alternative with Finnair and then using Avios on JAL from Seoul to Tokyo which is about a 1hr 20 flight. If you have Virgin Flying Club points, you can redeem these quite easily on ANA or Korean Air to and from Seoul to a range of Japanese airports. I have done Cathy Pacific to Tokyo via HKG, and despite the amazing lounges in HKG, it can be a long journey with the 4hr+ flight to and from Tokyo from HKG, unless you build in a few nights in HKG, of course. These routings via Seoul or HKG do need to be carefully planned, mind.

Other options which have worked for me over the years include flying from an EU airport (JAL from Frankfurt to Tokyo) or ignoring redemption and just looking out for some of the good fares available to Japan during their low tourist season. I can't really comment on BA as I no longer choose them to Japan because the ground and in-flight service for such a long sector is dated and inferior to JAL.