"How long does it take to switch to HDG mode or disconnect the autopilot and take over manually and stop an inappropriate turn?"
From my memory, they don't even noticed the aircraft was deviating from the course...The were in the rush (3 hours delay for takeoff at miami), and when they noticed the GPWS call, they called TOGA, but forgot the spoilers out....No altitude gained, hit the hill only 20 meters from the top....