Nuno has signed a three-year deal with the Hammers and will be in the dugout for their match against Everton on Monday.
West Ham are languishing in 19th after an awful start to the season under Potter, losing four of their five opening games.
The former Chelsea and Brighton boss had only been in charge for eight months, but was abruptly relieved of his duties on Saturday morning.
Nuno was himself sacked by Forest earlier this season after a rather public falling out with club owner Evangelos Marinakis.
The 51-year-old joins West Ham after guiding Forest to seventh in the Premier League last season, their highest finish since 1995.
"I am very pleased to be here and very proud to be representing West Ham," Nuno said in a statement.
"My objective is to work hard to get the very best from the team and ensure that we are as competitive as we possibly can be. The work has already started, and I am looking forward to the challenge that is ahead."