The 27-year-old Frenchman played 29 times for Leyton Orient in 2016-17, as they were relegated from League Two.
He was out of contract, having joined the O's from Portsmouth on an 18-month deal in January 2016.
Atangana made his name in England with Havant and Waterlooville, winning their player of the year in his solitary season at Westleigh Park before joining Pompey in 2014.
He is the third close-season signing made by Cheltenham boss Gary Johnson - who himself signed a new contract on Friday - after Yeovil midfielder Kevin Dawson and Dover defender Jamie Grimes.
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Al Hasawi, who took control of the Reds in the summer of 2012, would not reveal the identity of the buyers.
However, BBC Sport understands that a United States consortium, led by millionaire John Jay Moores, is on the verge of securing a £50m deal.
The takeover would see the former San Diego Padres owner take an 80% stake in the Championship club.
Moores' consortium were previously interested in buying Premier League side Everton, and the potential new owners met club staff last week.

The 57-year-old took over at Elland Road in April and is currently with the players on their pre-season tour of his native Italy.
"Next season I do not want one single player who lives outside of Leeds, no way. That is mandatory.
"Last year there were players who were coming late to games," Cellino told BBC Radio Leeds.
Cellino added that next season before a home game the players will stay together, have dinner together and go to games together.
"It is important to do this because the players must be able to sleep well the day before a game and maybe they cannot do that if they have small children or if their neighbour is having a party," he said.
"I found out that most of the games that are close to Leeds we go to by bus, but we don't have our own. Why not? I would love to have a Leeds bus and I am going to buy one."
Cellino is confident the club are close to bringing in more new signings following the £11m departure of striker Ross McCormack to Fulham.
The Championship side have brought in two players so far this summer, goalkeepers Stuart Taylor and Marco Silvestri.
Midfielder Tommasso Bianchi and defender Jonathan Rossini, both of Serie A club Sassuolo, are currently training with the team, while Catania striker Souleymane Doukara is also expected to join.
Leeds announced losses of £9.5m for the 2012-13 financial year and Cellino said the club had to "stop wasting money that isn't ours" when he took over.
Cellino, who claims that the club is now debt free, is now targeting a buyout of the 25% of the club still in the hands of previous owners GFH Capital as well as buying back Elland Road.
"There is no debt anymore and we have found a deal with GFH and I am glad for that," he added.
"We are close to raising £30m of capital and that is without the money we got for McCormack. We did not budget for his sale because we did not want him to go.
"The decision to buy back the stadium had been made before Ross was sold but now we can use that money for the deal. We want to take back our house."
Cheltenham Town have signed midfielder Nigel Atangana on a two-year deal.
[NEXT_CONCEPT]
Nottingham Forest owner and chairman Fawaz Al Hasawi says he is close to agreeing a deal to sell the club.
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Leeds United owner Massimo Cellino has ordered that all of the club's players must live in the city.