Listening to the recording, ATC had already pressed the crash alarm before the cockpit crew announced the evacuation to ATC and called for fire and rescue (which was also done in a very timely fashion). At which point you can already hear the EVAC wailer noise from the cockpit in the background.
There was around 12 seconds between the stop and the fire being mentioned in the mayday call, no doubt aviating before communicating, however I suspect the tower would have hit the crash alarm the moment smoke was spotted.