It's not the gamma ray burst that would do it. The gammas wouldn't
even reach the ground; air itself is sufficient shield.
What *would* do it is the relativistic particles that are expelled
from the collision. These are thought to be the source of the gamma
burst (by folks building models of bursters at cosmic distances.)
These particles would carry much more energy than the gammas,
and would arrive over a period of centuries (since the particles
would not travel in straight lines, due to interstellar magnetic
fields.) They would induce radioactivity in the atmosphere and
produce secondary particles that would reach the ground.
Paul