I don't know for sure, but the non-authoritative information I've found says that the 330 with the side-stick controller does NOT have a stick shaker.
The fly-by-wire system does have a de-facto stick "pusher" -- or more accurately, the flight control system will avoid or recover the plane from a stall as long as it is still working and you aren't close enough to the ground that it thinks you are landing.
I don't know what the autopilot does under those circumstances, once it recovers from the stall.
I located a chart of exactly what’s enabled and what isn’t for each operational state somewhere online. I’m trying to remember the search now...
Oh, here it is:
http://www.airbusdriver.net/airbus_fltlaws.htm
There’s reference to the pseudo-pusher there, too.
The details are so cobbled up at the moment, I’m at a loss to explain why there are so many “informed judgments” about what took place. Just today, I’ve heard the piece of data about speed described as being both over speed and under speed. And, of course, we now have word that the debris found wasn’t from the aircraft. I don’t think we’ll really know anything until the FDR is recovered.