It depends on various things.
BA Avios points are a poor value compared to AA miles if your goal is business class (or first class) longhaul international redemptions. BA Avios are a good value for shorter nonstop coach redemptions.
BA Avios points and AA miles don't expire as long as you have "activity" every 18 months of AA or every 36 months for BA.
By contrast, Qatar Privilege Q miles expire after 3 years (unless you are Qatar Platinum) and there's nothing you can do about it. (That's called "hard" expiration, and it's common for airline programs outside North America/UK.)
See the wikipost of this thread to compare mileage expiration policies at various airlines:
Other differences (which only apply if you fly enough to get "significant" elite status):
Upgrades tend to only be available within the program you belong to. But not all programs do upgrades the same, plus upgrades don't tend to be anywhere close to guaranteed (without higher status levels especially) so whether you could use them at all in any program is a question.
If you get to OW Sapphire, you get lounge access when flying internationally. That means you'd have more lounge access if your airline program is not in the country where you fly the most. So where you want to base for lounge access is sort opposite to where you want to base for upgrades. (But if you're not to get to the OW Sapphire level, it doesn't matter at all.)