There is no such thing (yet?) as an alliance-wide frequent flyer program. Splitting your flights over two or more programs in the same alliance is usually a bad idea, as the value of high-level status on one alliance member is usually greater than the value of lower status on two or more. (There are exceptions, especially if you can earn at least mid-level status on both, but they are exceptions.)
Several decision factors can influence your choice within an alliance. In no particular order:
1. How easy is it to gain elite status? Within oneWorld, for example, AA is better if you fly primarily on economy fares, BA is better (as regards this factor) if you fly a lot in paid first or business.
2. Which airline do you fly most? Generally, upgrades are reserved for that airline's own frequent flyers. Upgrades for frequent flyers of its alliance partners are harder to get or nonexistent.
3. Lounge access rules vary. If you fly around North America a lot, other than as part of overseas travel, the fact that AA elites don't get complimentary access to AA lounges but BA elites do may be significant. There are similar differences elsewhere.
4. The number of miles (or points, Avios, whatever) you need for a given award varies. Avios are distance-based, so they're great for short flights. They're less great for long ones, and BA is notorious for exorbitant "fuel surcharges." AA miles work out better for long flights. Can you plan ahead? In SkyTeam, Delta miles are good if you plan 11 months out and get an award in their initial allocation; once that's gone, and it tends to be skimpy, their mileage requirements double or triple to absurd levels.
5. Some airlines provide substantial benefits at the entry elite level. Others pretty much ignore them, saving the good stuff for second-tier and higher. You have to ask what elite level you're likely to reach in order to evaluate this factor.
Others will say more ...