From ICAO Doc 4444:
126.96.36.199 Start-up time procedures
188.8.131.52.2 Start-up time procedures should be implemented where necessary to avoid congestion and excessive delays on the manoeuvring area or when warranted by ATFM regulations. Start-up time procedures should be contained in local instructions, and should specify the criteria and conditions for determining when and how start-up times shall be calculated and issued to departing flights.
Procedures should be locally implemented. If no delay is anticipated by the ATC, the controller can issue "start-up at your own discretion" (184.108.40.206.4). Of course any delay on the pilot's part that will make them miss the takeoff time, they are supposed to report that delay. After all the ground crew, for example, won't wait forever.
For the start-up time procedures mentioned in Doc 4444, you need to check the AIP if such procedure is applied. For Bangkok (VTBD), for example, the AIP notes:
Starting up Procedures
Pilots are to call Don Mueang Ground on 121.9 MHz for push back and start up and should give parking stand number or location and ATIS information.
Unless other ATC restriction is imposed, the aircraft must be push back within 5 minutes from the time ATC clearance is received otherwise the ATC clearance will be cancelled.
Additionally, in order to provide a more flexible ground traffic movement, all domestic departures shall no longer be required to push back within 5 minutes after clearance received.
Which means the 5 minutes push back limit does not apply to domestic flights. The same document also lists the different procedures for each stand, for example for parking stand 21:
The aircraft may start one engine to idle power. They will be pushed back onto "taxilane B" to face either north or south behind the holding line, where remaining engines may be started. Breakaway thrust will be applied when cleared to taxi.
If unsure, always ask the ATC. If the push and start clearance is split, the ATC should specify that.