This is long and OT, so please forgive me...
Short answer to your last question is "not very good"... I've never been given an op-up when flying BA (my last BA flight was in full J with CW oversold--still no upgrade), but then again most of my transatlantic travel is on AA due to bennies and upgrades, and I don't do much business outside Europe and North America. That said, when I started spending more time in the UK a year or two ago I spent quite a bit of time looking at both BA and AA programs to determine whether one would do better than the other for me, and I decided to stick with AA.
To be honest, aside from the comparatively more difficult elite status hurdle on BA, the only two reasons for me to prefer BA over AA is elite recognition and MFU (miles for upgrade). On the former, BA does an comparatively
horrid job of integrating
oneworld elites into their own elite recognition. For example, a BA elite flying on AA automatically gets Priority AAccess treatment, which usually involves early boarding and premium security line access--AA lounge agents also are very good at knowing every possible legitimate way to access the lounge, and I've never heard of a BA Silver or Gold being turned away from an AA lounge. On the other hand, BA barely recognise the
oneworld levels at all, and many staff have no idea what they are or what they should entail. For example, BA lounges at Heathrow all show the proper oneworld level logos on entry signs, yet there are still incidents from time to time where AA elites who should get entrance are turned away. Same goes for check-in zones: T5 Zone J
should be open to all
oneworld Emeralds, yet I sometimes get a strange look when showing my AA EXP card to gain access to the area. That said, with both Zone J and Galleries First, while I've sometimes had to explain myself, I've never been turned away--but others on FT and elsewhere have... And I find it absolutely absurd that I have to explain something as well established as
oneworld elite benefits to BA's staff, when BA have been in the alliance for more than a decade. So, long story short, BA do a much better job of recognising their own elites, so if that kind of thing matters you may want to lean BA.
On the MFU front, the best way I find for using Avioseses is MFU, which is actually a great deal if you don't mind buying (or your company will pay for) WT+. For example, upgrading a return trip from economy to business across the pond on AA will run you 50,000 miles + US $700 copay. The same from WT+ to CW will run you 15,000-25,000 Avioseses with
no copay. The savings in miles and copay outlay on BA, combined with BA's (currently) superior business class product, makes MFU a
very attractive deal. BA also have pretty decent availability for booking straight into CW using MFU, which means less waitlisting than on AA. That said, SWUs do even things up quite a bit. Personally, what I do is earn some ancillary miles on BA (credit cards, Tesco, etc.) and use those Avioseses to MFU the occasional eastbound redeye on BA, then once the MFU is completed I switch the frequent flyer number to my AAdvantage one and earn there.
Overall, I think AA is right for me, and I am happy staying here. I tended to fly more paid business or first, I would probably switch to BAEC and shoot for Gold there.