The axes I'd animate on the wing controls of a hypothetical rig.
Describing these bone rotations can get a little confusing, I think it might help if I show a visual representation of them. By creating a hypothetical rig of a bird's wing we can see which axes we should animate to accurately mimic reality.
It might be preferable in production to create a IK wing rig but for simplicity I've shown a straight forward FK wing where all the bones in the wing would be represented an by an animation control.
I try to animate the Ulna and Radius control in just one axis, rotation y, as that is the only axis these bones can rotate in life.
The carpometacarpus' z rotation is what causes the main bend in the wing as the first joint rotates up in flight...
.... but by adding a little rotation z to this second control, you can create a curve at this point in the wing and stop the bend appearing as harsh. The first wrist joint above can rotate in y but this second joint cannot.
I animate the feathers along their length for drag, I usually find that just this rotation on the feathers is enough for an initial cycle, then I'll often add further rotations in the shot to break them up. Also, a sense of the feathers spreading on the down-stroke can be achieved by animating their base along the y axis.