Being (mostly) a Boeing devotee, I have very limited knowledge of how an A380 copes with a crossswind on takeoff. However, Trent 972 gives a clue to what happens from the FCOM quote. To paraphrase: 'With the sidestick neutral, ground law will try to keep the wings level'. There is only one way that this can be done, with neutral stick input, and that is by automatic application of aileron, rudder and possibly spoiler input, to eliminate or reduce roll and yaw.
Yaw will be sensed by the inertial systems and is a result of two effects: assymetric forces on the rudder ('weathercocking') or by excess lift on the 'into wind' swept wing which makes the aircraft want to roll, or on the ground, turn away downwind. I am fairly certain that the first effect is easily eliminated by rudder yaw damping , but the second effect is more subtle and would require an aileron input to minimise it.
However, in limiting crosswind conditions, the amount of aileron input that is sometimes required to keep wings level will cause the upwind spoilers to partially extend, increasing drag and reducing takeoff performance. That is why there is usually a limit to aileron input on takeoff and why, in limiting crosswinds, the aircraft wants to roll off downwind after rotation and you have to increase the into wind aileron input to keep the wings level. On the ground the aircraft's ability to roll downwind was prevented by the reaction of the downwind gear strut, but that restriction is removed as soon as the aircraft gets airborne.
The most elegant auto takeoff solution would be for the aircraft system to eliminate all yaw with rudder, sense gear strut compression and be allowed do what is neccessary with aileron to keep them equalised, without an excessive drag increase. Maybe that is what the A380 does already??