Müller started his karting career in 2004 where he finished sixth in the Bridgestone Cup Switzerland ICA Junior Class in the following year. He stuck around in the championship for the following year and improved to third before finishing runner up in the Swiss Junior Championship. In 2007, he won the Bridgestone Cup followed by fifth in the Swiss KF3 Championship.
He graduated to single seaters in 2008 when he competed in the Swiss Formula Renault 2.0 with Jenzer Motorsport. He claimed three podiums of which one was his first single seater win to collect 174 points and 5th in the standings. He then raced with the team in the Italian Formula Renault 2.0 for four races where he took 8 points to claim 32nd, the Formula Renault 2.0 WEC for two races to finish 52nd and the Italian Formula Renault 2.0 Winter Series for two races where he took a podium in each race to grab 52 points and 3rd in the standings.
From there, Müller stuck with the team in 2009 where he entered the Swiss Formula Renault 2.0 once again and the Eurocup Formula Renault 2.0. He took a single podium during the Eurocup to grab 25 points and 11th in the standings. For his second Swiss championship, he took 9 wins out of 12 races, taking a podium in the remaining 3 races to take 317 points and first in the championship. He then proceeded to move to Formula 3 when he stayed with Jenzer to compete in GP3 during 2010 and 2011. Müller took two wins and four podiums in his first year and finished third followed by fourth with 3 podiums and single win.
Headed into 2012, Müller moved back to Formula Renault by entering in to the 3.5 Series with International Draco Racing for two years. In his first year, he managed a single podium and 9th in the standings before taking 3 podiums of which 2 were wins in the following year to finish 5th in the standings. In 2014, he joined Audi Sport Team Rosberg to race in the Deutsche Tourenwagen Masters where he finished 19th. He also raced in a single race during the Blancpain Endurance Series with Audi Race Experience. Nico then returned with Audi Sport to race in the Deutsch Tourenwagen Masters in 2015 where he received more points however, only finihsed 21st. He also drove in the Blancpain Endurance Series again but with Belgium Audi Club Team WRT, taking a podium and 9th in the standings.
Müller started to join more series in 2016; repeating another year in the Deutsche Tourenwagen Masters with Audi Sport Team Abt for 18 races to take 2 podiums of which 1 was a win to place 9th, repeating the Blancpain GT Series Endurance Cup with Audi Sport Team WRT where he finished 10th with 1 podium, the Blancpain GT Series Sprint Cup with the team to finish 27th, the Intercontinental GT Challenge with Audi where he raced in one race and claimed a pole position, a fastest lap and a podium to finish 9th and the FIA GT World Cup with Phoenix Racing Asia but he did not start.
In 2017, 2018 and 2019, Müller raced with Audi Sport Team Abt Sportsline to respectively take 12th with 81 points and 2 podiums, 10th with 96 points and 2nd with 250 points. He also competed in the Blancpain GT Series Endurance Cup in all three years with Audi Sport Team WRT, earning himself 10th with 29 points, 39th with 8 points and 32nd with 3 points. Müller also joined Audi in competing in the Intercontinental GT Challenge where he finished with 8 points to place 12th, 12 points to place 18th and he remained out of the points for his final year.
Alongside the three races, in 2017 Nico also raced with Audi Sport Team WRT for a single race in the FIA GT World Cup but did not finish, the World Rallycross Championship with EKS RX for two races where he finished 17th, FIA World Endurance Championship LMP2 with G-Drive Racing for a single race where he placed 27th. Müller also entered the World Rallycross Championship with EKS Audi Sport in 2018 for a single race and finished 19th. Other than that, he competed with Ipiranga Racing in the Stock Car Brasil in 2018.