Taylor, 31, has signed a two-year contract at Valley Parade following his release from Charlton, while Nelson has returned to Scotland with Hibs for an undisclosed fee after joining Bradford from Kilmarnock in January.
Bantams boss Phil Parkinson said: "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.
"We have a tight-knit group here, so I couldn't afford to give someone up if I wasn't able to bring someone else in.
"Nelse came in last season and played his part in helping the side into the Play-Offs, and we thank him for that."
Taylor, who began his career at Exeter, made 14 appearances for Charlton in the Championship last season and Nelson, 33, played the same number of games for Bradford.