If you tend to fly NH to Japan and UA/US/BD transatlantic, it would be worth your while looking into BD: the elite levels are not too onerous (16K and 38K) - you are UK based, so you could earn miles on the BD Mastercard - ANA fares normally earn full mileage on BD (and double mileage in C) and redemption is very good with the miles+cash option.
Not sure about comping with JAL. BD are not averse to comping, but they would probably not regard JAL as a competitor and would not necessarily feel incline to comp (you can always ask them, though)

For comping, your best bet tends to be North American airlines. Among them, AC would be the one to go if you want an easy path the elite qualification and UA would be the one to go for more substantial/usable benefits since you do not seem to fly much to Canada. US Airways is also a possibility, although it depends on how long you think they will survive.