An existential emergency is just that — the sudden emergence of a personal situation whose circumstances seem to threaten personal existence. An epistemological emergency is something else — the sudden emergence of a situation (whether personal or not) whose circumstances require the drastic interrogation of those circumstances with questions like, "How do you know???"
These latter cases are wearisome but seem to be ever more common these days as the date for submission of the epistemological arguments comes near. I have to suppress my urge to reply, "Yes, well, think of this: HOW do you know, how DO you know, how do YOU know, how do you KNOW?" and provide more obvious scaffolding (no, not the type with a trapdoor and gibbet).
Thank God I have not yet had to cope with any of my students' genuine existential emergencies. Yet.