Astonishingly, I believe the first cockpit indications of a problem were probably inhibited by the EICAS (or Airbus equivalent) until the aircraft was above the usual 400ft agl,
No idea how the Airbus logic works, but on Boeing EICAS the takeoff inhibit doesn't kick in until 80 knots, so if there is something wrong you can do a low speed abort.
On Boeing, if the FADEC detects a serious fault, EICAS message "ENG X CONTROL" (L/R ENG CONTROL on twins) is displayed - the procedure is No Dispatch. ENG CONTROL is inhibited above 80 knots and in flight - the logic being there is not procedure once airborne, and if the engine is still running we don't want the crew to shut it down because of the message.