Folk last kicked in the NFL with the Tampa Bay Buccaneers in 2017.
Nugent's release comes after a rain-soaked 27-13 win over the Cleveland Browns on Sunday in which he missed a 34-yard field goal, and a 29-yarder was blocked. It was the first blocked field goal against the Patriots since 2006.
"I thought we needed to make a change," Patriots coach Bill Belichick said Wednesday.
Of Folk, Belichick said: "We'll see how it goes. Haven't seen much so far. He's had a very successful career, but we'll see where we're at."
Belichick said Folk's recent workout with the Patriots, more so than his experience in the Alliance of American Football, was a significant factor in the decision to sign him.
"I'd say that was more important than the World League, or whatever it is," Belichick said, referencing the NFL's former "minor league," which became NFL Europe, that operated from 1991 to 2007.
Nugent, 37, had been signed on Oct. 3 after the Patriots placed longtime kicker Stephen Gostkowski on season-ending injured reserve with a hip injury. Nugent appeared in four games, and was 5-of-8 on field goals and 15-of-16 on point-after attempts.
The Patriots are 8-0, but with the competition level of their schedule rising and the potential for closer games ahead, the instability at kicker is a potential concern for coach Bill Belichick, who has had the luxury of having Adam Vinatieri and Gostkowski as his two kickers over most of his 20-year tenure.
Folk, who also kicked for the Dallas Cowboys (2007-2009) and New York Jets (2010-2016), is 245-of-305 on field-goal attempts in his career. Earlier in 2019, he was the kicker for the Arizona Hotshots of the Alliance of American Football.
Since Gostkowski went on IR, the Patriots have brought in kickers to keep their emergency list updated. The group included veterans Folk, Blair Walsh, Giorgio Tavecchio and Kai Forbath, and younger options Greg Joseph, Austin MacGinnis, Austin Rehkow, Nick Rose, Matthew Wright, Josh Gable and Elliott Fry.
The Patriots also released tight end Eric Tomlinson, who started two games and had one reception for one yard since being signed on Oct. 15.