As a tower ATCO I’d maybe know there’s an issue before you tell me or deviate from the SID just based on what the aircraft is doing, but aviate/navigate/communicate is always expected. If PNF has a moment, a quick “Mayday c/s engine fire - Standby” sort of call could be helpful to highlight potential separation issues, clear the runway and get the emergency services there more quickly, but it’s quite likely all of that would be happening even if you said nothing.