Do pilots on approach receive automated warnings of sudden windshear or seismic motion?
Windshear, yes. Seismic events, no.

Most major U.S. airports have a LLWAS -- Low-level Windshear Alert System, which is essentially an array of anemometers around the airport which detect wind speed and direction.  The values from each station are processed by a computer that uses algorithms to detect a windshear or microburst event, and it notifies controllers in the tower.  As a simple example, if the anemometers around the perimeter of the airport are all pointing toward the center of the airport, that's a strong indication that there is a microburst over the center of the airport, with air descending into the ground and then spreading out in all directions.  Obviously, the algorithms can detect cases beyond just that.

And, many airports supplement that system with Doppler radar as well.

Regardless of how they're generated (LLWAS with anemometers, Doppler, pilot reports, visual observations of blowing dust, etc), the warnings are broadcast by the tower controllers, rather than by any automatic system.  In the aircraft, aural warnings from the GPWS (reactive windshear -- noting changes in groundspeed and sink) and the EGPWS (predictive windshear -- returns from the aircraft weather radar) are automatic, but they don't block a radio frequency.  Automated alerts over the tower frequency might block an even more important transmission, so it's left to the controller's judgement when to broadcast the alerts.  (They're usually pretty prompt!)

No airport I'm aware of has a system similar to what the OP described for seismic events.  I would assume that the risks to a high-speed train from an imminent earthquake -- i.e. the train may derail at high speed with catastrophic results -- are far greater than the risks to aircraft.  I've never read anything about what happens when you're landing or taking off in an earthquake, but I'd doubt that anything other than the most severe would have much effect.  And if the earth is shifting enough that you can't keep the aircraft on the runway, well, it's a really bad day for pretty much everybody in that state that day!