The 19-year-old, who joins for an undisclosed fee, is the Canaries' third signing of the summer following the arrivals of Ricky van Wolfswinkel and Javier Garrido.
Speaking on his move to the Barclays Premier League side, Redmond told canaries.co.uk: "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. The people around the football club and in the city that I have spoken to so far have attracted me to this club so much, it's made me realise just how much this club means to the people of this city."
Redmond came through the Birmingham Academy and played under Hughton when he was manager at St Andrew's, which included a run in the Europa League for the Championship club.
The winger, who featured for the England under-21s in this summer's European Championship, is looking forward to linking up with his old manager again, this time in the top flight.
"He 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," Redmond added.
"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."
Birmingham are understood to have also agreed a fee, in the region of £2million with add-ons, with Swansea, but the winger opted instead to move to East Anglia and reunite with his former manager.
Hughton said: "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 is 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 is very keen to develop his own game, is 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 am sure the move won't faze him, but this is still a part of his development and one he is very much looking forward to."
Hughton is keen to further bolster his squad ahead of their three-match pre-season tour of the United States, and has been linked with a swoop for Sweden international Ola Toivonen at PSV Eindhoven.
However, earlier this week the Norwich manager maintained the club would resist any moves to lure away England goalkeeper John Ruddy, with Chelsea ready to significantly improve their £5million offer which was swiftly rejected.