Nuno was sacked in the early hours of Tuesday morning following a breakdown in relations with club leadership and a difficult start to the season.
Postecoglou, 60, takes charge with immediate effect and will be in the dugout for Forest's Premier League clash against Arsenal on Saturday. He is expected to be joined by several members of his former Tottenham coaching staff.
The Australian coach has been out of management since parting ways with Spurs in June, just weeks after delivering the club's first major trophy in 17 years - a dramatic Europa League final victory over Manchester United. Despite that continental success, Tottenham finished a disappointing 17th in the league, with 22 losses - the most ever recorded by a team that avoided relegation in a 38-game Premier League season.
Forest owner Evangelos Marinakis praised the appointment as a significant step forward in the club's journey.
"We are bringing a coach to the club who has a proven and consistent record of winning trophies," said Marinakis. "His experience of coaching teams at the highest level, along with his desire to build something special with us at Forest, makes him a fantastic person to help us on our journey and achieve consistently all our ambitions.
"Ange has the credentials and the track-record to do this, and we are excited he is joining us on our ambitious journey."
Arsenal have been handed a timely lift ahead of their Premier League clash with Nottingham Forest on Saturday, with several players returning to training at London Colney.
Ange Postecoglou has vowed to restore Nottingham Forest to the top level of English football after being confirmed as the club's new manager.
Former Manchester United midfielder Christian Eriksen has completed a move to Bundesliga side VfL Wolfsburg on a free transfer, signing a contract until 2027.
West Ham have confirmed the return of veteran goalkeeper Łukasz Fabianski on a contract that runs until the end of the season.
