The move reunites Redmond with former Blues boss Chris Hughton, with the England U21 international signing a four-year deal at Carrow Road.
He was one of the bright spots of an otherwise disappointing European Under-21 Championship campaign for the England side this summer.
The 19-year-old follows defender Curtis Davies out of the door at St Andrew's after the defender joined Hull City for £2.25m last week.
Redmond told his new club's official website: "I'm very happy. It looks like a fantastic club, I'm just looking forward to getting started now.
"I know that there is only one football club in this city and it has an excellent fan base.
"[Chris Hughton] was very influential on my career at Birmingham. He gave me the chance to play and the chance to express myself every time that I stepped on to the field and I feel like on the bigger stage he will do exactly the same thing.
"I would probably say I have played some of my best football under him so far. Last season was a good one for me in terms of goals and assists and just general performances.
"Hopefully I can continue the form that I left off with last season at Birmingham."
And Hughton added: "Nathan is a young and exciting player that I had the pleasure of working with at Birmingham.
"He is a player that was of interest to us for some time and the situation has now come about that we can get him on our terms.
"He's someone we can bring in to the forward areas and primarily has played on either wing in his career so far, but can also play off the front.
"He's very keen to develop his own game, he's one of these lads that likes to be on the training pitch and is usually one of the last ones off the training pitch.
"I'm sure the move won't phase him but this is still a part of his development and one he is very much looking forward to."