I push my GLA pretty hard in all weathers and I have never heard any strange noises from the wheels when in motion. What you describe could be down to the tyres breaking grip, as if it was a mechanical problem you'd hear it more in the dry when the forces acting on the wheels and bearings etc. are higher. Are you running winter tyres? If not, maybe you should be.
The only other thing I can think of is that you are losing grip on a wheel (due to the wrong tyres for the conditions), which is then spinning and having the brake on that wheel automatically applied by the traction control system. I get a similar sound when pulling away from rest on a steep incline, which has to be happening at the moment that the brakes are automatically released, as the wheels pick-up drive from the engine. The systems that control the traction are very complex and the GLA has huge brake disks, so some noise is inevitable when the wheels are braked or released by the car's various systems.
