Do airlines have an ATC schedule they have to adhere to?
The estimated off block time, EOBT, which is the time at which the flight is planned to leave the gate at the departure airport, has a validity of +/-15 minutes. If the plane is not able to depart within this window, the company must send a new ATC flightplan with an updated EOBT.

In some cases a calculated takeoff time, CTOT, is issued to a flight. This is the exact time the plane must begin its takeoff roll. It is given based on airspace or airport congestion along the planned route of the flight, to prevent overload in individual ATC sectors. The flight must begin its takeoff roll within -5/+10 minutes of the CTOT. If unable to comply, a new CTOT must be requested.