With 20+ seats, you will probably likely run out of saver awards on some routes. This is why you have to plan on redeeming a portion using anytime awards (twice as expensive and requires AA miles, but very available).

First, try to redeem as many saver economy awards using BA miles. Then, put on hold as many saver awards as you can using AA miles (probably business/first awards at 37.5K each way / 75K round trip, since your earlier BA redemption would have used up all the saver economy seats).

Once you are done, call AA and ask for award availability for your remaining passengers on either HA or AS. Try to see if there are ways to book the remaining seats using those two partner airlines, then put the tickets on hold.

(note that AS is also a BA partner, so technically you can phone BA and book them earlier using BA miles, if you want).

Finally, hold all remaining seats using the doubly-expensive AA anytime awards.

Review your bookings and ticket as needed. For AA-only ticket, you can redeem online. For partner tickets, you will have to call in and pay the phone ticketing fee.

In your case, you absolutely should get a subscription to Expert Flyer, at least for a month or two. They allow you to directly see the precise inventory numbers for award seats (up to 7 at a time), and would save you much time from having to try to figure out how many seats to try to hold for.

Actually, I would also highly recommend that you consider hiring a mileage expert. They usually charge $200 - $400 for a few passengers, but you may be able to negotiate something for your 20+ bookings. The hire can easily pay for itself, if by their knowledge and knowing how to redeem, they manage to save you just 1 or 2 extra saver awards.