"WITHOUT DELAY: Emergency descent accomplish"
ATC coordination comes later. Aviate first.
Don't know where you aviate but certainly on the North Atlantic Tracks, even using Strategic Offset you can end up with lots of traffic below you, all same way, same day and in very close proximity.
I'd say that's one instance where "Aviating first" might mean well mean
delaying the decent for a few seconds to turn off track before initiating descent.