Anything you want can happen. The Alcubierre bubble isn't a thing that appears through ordinary physical cause and effect. It's a spacetime geometry that Alcubierre wrote down by fiat, computed the stress-energy tensor field of (which you can do for any geometry), noticed that it didn't match any known physics, and published anyway. So pick whatever you want the colliding bubbles to do, compute the stress-energy tensor field for that, use your sufficiently advanced magic to make that stress-energy tensor field happen, and you're golden. There's no physical principle selecting a particular outcome because the original solution was never constrained by any physical laws.
Relatedly, there's really no such thing as an Alcubierre "drive". The configuration of matter necessary for the bubble to exist can't come from the thing being transported, because the former is outside the latter's future light cone. You need to start setting the bubble up well ahead of time (~N years ahead of time for an N-light-year journey), unless you have some way of locally (special-relativistically) exceeding the speed of light. See Alcubierre drive#Placement of matter in Wikipedia.