http://www.oneworld.com/ffp/extra-baggage-allowance seems to indicate you might be out of luck as the extra baggage features for OneWorld Emerald (US Chairman/AA EXP) and Sapphire (US Gold/Platinum, AA Plat) have the caveat that "This benefit is not available on British Airways 'Hand Baggage Only' fares on select short-haul routes."

I've been flying tons of both short and long haul BA in the past few months as a Chairman (OW Emerald). I never check bags so can't give you much intel in that space, but as Sapphire you'll get priority boarding and the ability to use the business class lounges as well as business class checkin. Don't expect more than that - not sure if you get "FastTrack" for security at LHR/LGW as a Sapphire in coach, but I think it's a decent possibility. BA's onboard hard product is pretty weak even compared to other legacy European carriers, both in long haul business and certainly on short haul "Club Europe". Their long haul business seats are not remarkable and a far inferior seat to, for instance, the seats on the US A330 fleet, even on BA's new A380s. My travel patterns have forced me to use BA or otherwise not earn any US miles as they've been the only OW carrier on the routes I've needed, but when I've had the option to use other OW carriers - CX and QF in the past week for instance - it's been pretty remarkable how far behind BA is. BA's service on board is usually pretty decent though.