I assume it is just a rename of Wimp Out, but more suiting to a badass mechabug.
Alright, here's the leaked description of the Ability (I think):
"The Pokemon switches out when its HP becomes half or less."
Compared to Wimp Out:
"When this Pokémon's HP drops below half in battle, it will run away or swap out for another Pokémon."
Now look at the gameplay footage of Wimp Out:
It seems pretty obvious that Wimp Out activates as soon as HP drops below half, similar to Eject Button. Based on the assumption that Emergency Exit is just a more dignified name for Wimp Out, the logical assumption would be that it works the exact same way,, and that Emergency Exit activates as soon as Golisopod is attacked.
On top of that, if it activated at the end of the turn, I feel as though it would be mentioned in the description. Perhaps something along the lines of:
"When the Pokémon's HP becomes half or less, it takes one last swing and then makes a hasty retreat."
However, nothing like this is mentioned in the description of Emergency Exit, and that would be a pretty big feature to leave out, ESPECIALLY because it appears to be a reskin of Wimp Out. If there was a difference between the two Abilities, it would be mentioned in the description.
TL;DR if it were any different than Wimp Out there would probably be something in the description, but I can't say for certain because I have not seen the Ability in action. My answer is logically speculative.