Nyck started karting in 2004, taking his first championship podium in 2006 when he placed second in the Cadet Belgian Championship. He also took third in the ICA Junior Dutch Championship in 2006. De Vries competed in five different championships across 2007 where he placed second in every single one; KF3 Dutch KNAF, Junior Chorono Dutch Rotax Max Challenge, Junior Chrono Rotax Max Winter Cup, Belgium Junior Rotax Max Challenge and the KF3 Belgian Championship. He took his first karting win in 2008 in the KF3 WSK International Series and the German Junior Kart Championship. He also took a win in the 2009 German Junior Kart Championship, the KF3 CIK-FIA European Championship and the SKF WSK International Series. Nyck then took second in the 2009 KF3 20th Trofeo Andrea Margutti. He remained strong throughout the rest of his karting career; winning the 2010 CIK-FIA World Karting Championship, 2011 CIK-FIA World Championship and the 2011 KF1 WSK Euro Series, second in the 2010 SKF WSK World Series and 2010 SKF WSK Euro Series and third in the Super KF & KF1 CIK-FIA World Cup.
He made his debut in single seaters in 2012 with R-ace GP in the Eurocup Formula Renault 2.0 and the Formula Renault 2.0 NEC. He took two podiums in the Eurocup to finish 5th and four in the NEC, of which one was a win, to finish 10th. He continued in Eurocup but made the move to Koiranen GP for 2013. Nyck also joined the team for 6 races in the Formula Renault 2.0 Alps. He took 5 podiums, of which 2 were wins, to finish fifth once again in the Eurocup. Nyck also took 2 podiums in the Alps to finish 8th. For 2014, he stuck with the team for both championships. In the Eurocup, he took 10 podiums with 5 of them being wins and in the Alps he took 12 podiums with 9 of them being wins to finish first in both championships.
De Vries graduated from both series to the Formula Renault 3.5 Series with DAMS for 2015. Out of 17 races, he took a single win and 5 more podiums to finish 3rd in the standings. He moved to ART Grand Prix for the GP3 Series in 2016 where he finished 6th after two wins and 3 other podiums. Nyck graduated from there to the FIA Formula 2 Championship in 2017 with Rapax for 13 races and Racing Engineering for 8 races. In total, he took 1 win and 5 podiums in his rookie year to lace him 7th in the standings. He moved to Pertamina Prema Theodore Racing for a full season in his second year where he took six podiums, of which 3 were wins, and six fastest laps to finish 4th in the standings. His final year in the FIA Formula 2 Championship was in 2019 with ART Grand Prix where he took 12 podiums, of which four were wins, and 5 poles to win the championship.
Between 2018 and 2019, Nyck raced with Racing Team Nederland in the LMP2 FIA World Endurance Championship for six races where he took 2 fastest laps and finished 9th. He also currently competes in the championship with the team for the ongoing 2019 to 2020 season where he has taken a single win out of his four races and sits at 9th in the standings.
In 2019, Nyck joined Racing Team Nederland to race in the LMP2 24 Hours of Le Mans where he finished 15th.