Disclaimer: I haven't watched the docudrama.
For a comet impact, I think the chemical effects (dumping poisonous gases into the atmosphere) would be completely overwhelmed by the thermal effects (lots and lots of kinetic energy converted to heat).
Those poisonous gases would surely be converted to plasma and/or blasted out in the shockwave from the impact. Any that were still solid at the point of ground impact would be splashed out in all directions, where they would quickly mix into the (suddenly superheated) atmosphere. Carbon monoxide, methane, and ammonia could burn once they COOLED enough to combine with atmospheric oxygen, but they'd already be diluted below toxic levels.
The heat from the impact would also cause all these products to rise into the stratosphere as a giant mushroom cloud, drawing in fresh air at its base.
So, no, I don't see how this could be realistic. I'm almost tempted to go look for the video to see just how bad it was... but not quite.