Collins has signed a six-year contract at the G-Tech Community Stadium.
The Republic of Ireland international spent just a season at Wolves after joining from Burnley and made 26 Premier League appearances last season.
Collins is Brentford’s third signing this summer after goalkeeper Mark Flekken and Kevin Schade both joined from Freiburg.
Brentford’s previous record buy was the £20m paid to bring Schade permanently from Germany after his loan spell.
Speaking after the announcement of the transfer, Brentford manager Thomas Frank told the club's official website: "I'm very pleased that we've managed to sign Nathan. He's a very talented centre-back and still young.
"He has a lot of very good abilities that we value a lot. He's a composed and calm defender. His heading in both boxes is a big thing, both in open play and from set-pieces, which is massive in the Premier League.
"I also like his ability on the ball. He can carry the ball out of defence and also has the composure to find the right pass during build-up play.
"His character is really good, and we know that character is everything. He will fit into our culture, I have no doubt about that. I see leadership potential in Nathan. He's a perfect Brentford player - hungry and ambitious with a desire to learn."
Collins added: “I talked with the manager and a few of the backroom staff about the project here.”
“It’s a club on the rise and I can see where they want to go. It’s exciting to be here.
“I like to play football. I want to dribble, play and create. It starts at the back. I can play long balls and short balls. But, at the end of the day, I’m a defender and I have to head the ball, win it and win my challenges.”
