On my type the earliest we can engage an autopilot is the later of 5 seconds after lift off or 100′ agl. Same applies for all engines working or in the event of an engine fire/fail.
I always chuckle when I see people thinking they are really clever ‘hand flying’ the aircraft for an age after take off when all they are doing is following the flight director bars. For me, hand flying is something I do on arrival in reasonable visual conditions on raw data (flight directors off) using basic piloting skills.