O'Driscoll, 55, was appointed as successor to Steve Evans at Crawley in May after spending five months at Forest as assistant to Steve Cotterill, but has opted for a swift return to the City Ground as manager following the takeover of the East Midlands club by the Kuwaiti-based Al Hasawi family.
O'Driscoll said: "I would just like to take the opportunity to thank everyone at Crawley Town.
"I have been made very welcome by the staff and players and was looking forward to the new challenge of establishing the club in League One.
"As you can appreciate, opportunities like this one with Nottingham Forest don't come along too often. I appreciate Crawley Town's integrity and professionalism in dealing with the matter and allowing me to take up this new challenge.
"I wish Crawley Town every success for the future and I'm sure the club will continue to go from strength to strength."