The Airbus managed speed protection system is known as 'GS Mini'. It adjusts your bugged speed to achieve a minimum groundspeed. The autothrust (if engaged) foolows the command. However, the Airbus doesn't always cope (esp. a heavy A321) and situations like the one you describe happening to the 737 aren't uncommon in the bus.
The windshear function is a part of the GPWS and will tell you that you are encountering it - i.e you have to have it to get the warning. The Predictive Windshear System does use the Wx Radar, but you have to have moisture in the air for the radar to detect it. This is a common misconception and I often hear Pilots I fly with talk about their reaction to a PWS event during the deaprture brief when there isn't a cloud in the sky!