The deflection of the aileron doesn’t increase its own AOA or that of the wing (due to the way that AOA is defined), but the critical AOA of a flapped wing is reduced, so a wing with a downward deflected aileron may indeed stall before the unflapped one. This is often described as the mechanism behind a cross-controlled stall.
This is a good model, but leads to a surprising conclusion.
Looking at Fig 100 in Abbott and von Doenhoff, looks like a 0.2c plain flap indeed reduces the stall AoA by about a degree per 12 degrees of flap. But the interesting points arise if you compare a flap deflection of 0 at an AoA of 14 degrees with a higher flap deflection (say 40 degrees) at the same AoA. The wing without flap deflection is not stalled. The wing with flap deflected is stalled -- indeed the stall (Clmax) AoA has reduced -- but it still produces more lift than the unflapped wing! It's very difficult to find a counterexample where the lift goes down when flap is deflected, except at very high flap deflections (e.g. 60 degrees) or beyond the stall of the unflapped aerofoil.
So your ailerons still work, even close to the stall, albeit less effectively.