The 31-year-old has put pen to paper on a two-year contract at Valley Parade.
Bradford manager Phil Parkinson told reporters: "Matt is a good signing for us. He played a key role in the Charlton side that got promoted to the Championship a couple of seasons ago and was part of a successful Exeter side before that.
"He's a good age for a centre-half and he's a winner. I'm sure the supporters will enjoy watching him play because he gives it his all.
"Nelse had the opportunity to go back up to Scotland, but I needed both deals to work before either one could go through."
Bradford will begin their League One campaign on Saturday with a trip to Bristol City.