Wirelessly posted (BlackBerry8530/5.0.0.601 Profile/MIDP-2.1 Configuration/CLDC-1.1 VendorID/417)
MileSAAver and AAnytime awards are features of AA's program, not BA's. When you redeem BA miles for travel on AA, the award inventory comes out of MileSAAver inventory.
BA only offers capacity-controlled awards (like AA's MileSAAvers), even on BA flights. Additional award-seat availability may be made offered on BA flights to those with status in BA's FFP.
Also, note that when you redeem BA miles for travel on AA, BA's award chart and award rules apply, not AA's.