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."
Nottingham Forest have parted ways with Nuno Espirito Santo, with the Portuguese coach sacked early on Tuesday after tensions with owner Evangelos Marinakis reached breaking point.
Hugo Ekitike has embraced the high-stakes competition at Liverpool following the club's British-record 125 million pound signing of Alexander Isak from Newcastle.
Arsenal will resume their Premier League campaign on Saturday when Mikel Arteta's side host Nottingham Forest in a 12:30pm kick-off at Emirates Stadium.
Bayern Munich's honorary president has firmly dismissed the possibility of Nicolas Jackson making a permanent move from Chelsea, stating there's no way the striker will fulfill the required conditions for the transfer.
