The England U21 international joins Saints in a permanent transfer, signing a four-year contract.
Wood said: “It’s obviously a massive club, a Premier League club that was in the Championship last season, and to get straight back into the Prem is a big thing. I’m very happy to be here.
“It’s always been an attractive club. Obviously the style of football is a big thing, the coaching is a big thing, and it’s a step up in leagues for me as well.
“It’s a no-brainer really when you put all those things together. It’s quite an easy decision to make.”
Saints boss Russell Martin also told the club's website: “Nathan is a player I know and trust from our time together at Swansea, and someone I also tried to sign in my first job at MK Dons. He’s a top defender with tremendous physical qualities, great pace and calmness on the ball.
“What’s equally important to us is that he’s a great character who will settle in seamlessly with the brilliant group we already have here. He’s hungry to prove himself in the Premier League, and we want to help him achieve that.”