Shelvey, 30, has made only three Premier League appearances this season and his contract expires in the summer.
He has joined on a two-and-a-half-year contract, while Felipe, 33, has signed until June 2024.
Navas, 36, will provide cover for the rest of the season after Dean Henderson sustained a thigh injury.
Former England player Shelvey, two-cap Brazil international Felipe and Costa Rica stalwart Navas are manager Steve Cooper’s fourth, fifth and sixth signings of the January transfer window.
Shelvey said he had heard good things about Cooper, and added meeting the 43-year-old has been a “breath of fresh air”.
He told the club’s website: “It’s a pleasure to be here. As soon as I found out Forest were interested, it caught my eye straight away and I just couldn’t wait to get down here.
“It’s a fantastic football club that has so much history and I’m really looking forward to getting started.
“[Cooper] told me what he expects and the high standards he sets, so I hope I can come and help the team push to stay in the Premier League.”
Shelvey joined Newcastle from Swansea City in 2016 and made 202 appearances, scoring 18 goals.
Felipe, who has found his playing time limited at Atletico this season, said it was a “dream” to play in the Premier League.
“I’m really excited to be part of this wonderful, fast-growing club,” he added.
“You can see it’s a club looking to really grow. It’s a big challenge for me and I made a clear decision to be part of the history here and grow with the club.”
Navas, meanwhile, spent five seasons with Real Madrid, winning the Champions League three times and La Liga before moving to PSG in 2019.
Forest have also signed Newcastle striker Chris Wood on loan this month, bought Palmeiras midfielder Danilo for £16m and recruited his team-mate Gustavo Scarpa on a free transfer.