How close does a plane have to come to the ground before triggering the ground collision alert?
There are 7 modes of warning

Mode 1. Large descent rate near terrain. Gear and flap position not a factor in the warning.

Mode 2. Large closure rate with terrain.

Mode 2A. Occurs for a large closure rate if the flaps are not in the
landing configuration and glideslope deviation is more than 2
dots.

Mode 2B. Alerts for a large closure rate if the flaps are in
the landing configuration.

Mode 3. Alerts for a large altitude loss during takeoff or
during a missed approach with the flaps not in the landing
configuration or landing gear not down.

Mode 4. Alerts when the airplane is too close to the
terrain and the landing gear or flaps are not in the landing
configuration

Mode 4A alert when the landing gear is not
down and gives the aural messages TOO LOW GEAR at
low airspeeds, or TOO LOW TERRAIN at high airspeeds.

Mode 4B alerts when the landing gear is down and the flaps are not in landing configuration and gives the aural messages TOO LOW FLAPS if the airplane is at low airspeed, or TOO LOW TERRAIN at high airspeeds.

Mode 5 alerts when the airplane goes below
the glideslope during approach if the landing gear is down.

Mode 6 gives aural callouts when the airplane descends through set altitudes with the landing gear down. It gives callouts like MINIMUMS or DECISION HEIGHT.

Mode 7 gives windshear warnings.

Every one of these modes is altitude, speed and aircraft configuration dependent.

In the Enhanced GPWS, these triggers are also changed based on the position of the aircraft.

Regulatory requirement 14 CFR 121.354

Technical requirements TSO-151C