The Republic of Ireland international travelled to the Midlands for a medical on Tuesday after talks accelerated late on Monday night.
Both clubs agreed a £20.5million fee and Collins will travel with the rest of the squad to their pre-season camp in Spain. He has signed a five-year deal, with the option of a further 12 months.
The 21-year-old moved to Burnley last summer in a £12million deal from Stoke, which could have risen to upwards of £15million with add-ons. The Irishman came through at Dublin-based club Cherry Orchard before moving to Stoke as a youngster.
The right-sided centre-back also has some experience playing at right-back and fits the bill for Bruno Lage, who was targeting a right-footed defender.
Collins has made six appearances, and scored for goal, for his country after making his senior debut in October last year.
Last season for Burnley, in his first Premier League campaign, he made 19 appearances and scored two goals.
The signing is seen has somewhat of a coup for Wolves, as the defender is highly rated among fellow Premier League clubs.
Technical director Scott Sellars said: “We’ve looked at Nathan for a while and admired him, so we’re delighted to bring him to the football club. He’s a very talented young man – he’s only 21, has played international football and had a good season last year with Burnley. We think we’ve signed a player with a lot of potential, as well as being a very good footballer now.
“We’re getting someone who is physically very good, very tall, very athletic, and is very comfortable on the ball – he can step in and take the ball forward, which is what we need. He’s very good in both boxes and has real aerial strength.
“The feedback on his personality is about his leadership qualities, he’s a good character, so he ticks all the boxes. He comes across as expected, a humble young man who is appreciative of the opportunity, so we’re all going to be very happy.”