Sims started his karting career 1999 where he went on to take his first win in 2000 when he won the Comer Cadet Kartmasters British Grand Prix. He also took first that same year in the Super 1 National Comer Cadet Championship. Sims progressed quickly in karting, going on to take first in the 2002 ICA Junior Kartmasters British GP, 2002 Super 1 National JICA Championship, 2002 ICA Junior Monaco Kart Cup, 2003 Super Libre Kartmasters British GP and the 2004 Formula A 34 Degree Torneo Industrie. Alongside that, he also took second in the 2006 ICA Florida Winter Tour.
In 2006, Alexander made the move to the Formula Renault UK Winter Cup with Manor Competition for four races. He took a single podium and finished 9th in the standings for his rookie year. He returned for a second year in 2007 with the team where he raced in four more races at the Winter Cup where he managed to finish 19th. He also enter Formula Renault UK with the team where he managed a single win and two further podiums to finish 8th in the standings. Sims also stuck with the team for the Formula Renault 2.0 NEC for two races where he finished 37th in the standings. Alexander also joined Pole Services for four races and Manor Competition for five races in the French Formula Renault 2.0 where he took pole with Manor and placed 15th in the standings. For 2008, he returned with Manor Competition for twenty races in the Formula Renault UK where he managed two wins and a further ten podiums to finish second in the standings . He also joined the team as a guest for two races in the Formula Renault 2.0 WEC. Sims raced in the Eurocup Formula Renault 2.0 series with SG Formula for 2 races to finish 19th and as a guest with Team Meritus in the Formula BMW Pacific.
Heading into 2009, Sims made the move to Formula 3 where he joined Mücke Motorsport in the Formula 3 Euro Series. Out of his 20 races, he took one win and four podiums to finish 4th. He also raced with the team in the Masters of Formula 3 where he finished 5th. Alongside, he joined T.S.P. in the International Formula Master as a guest. He finished off his year by entering the Macau Grand Prix with City of Dreams / Räikkönen Robertson where he placed 18th. Sims returned back to Macau with Räikkönen Robertson Racing in 2010 but didn’t place. Alongside this, he raced in the Auto GP with Charouz-Gravity Racing where he placed 25th. Full time, he joined ART Grand Prix in the Formula 3 Euro Series where he took a single win and four podiums to finish 4th. Sims also joined the team as a guest in the British Formula 3 Championship and in the Master of Formula 3 where he finished second. Alexander moved to the GP3 Series with Status Grand Prix in 2011 where he finished 6th by taking a win and four podiums. He also entered in to the British Formula 3 Championship with Motopark where he took a single win and finished 18th and the Macau Grand Prix with TOM’S where he finished 18th. He went on to race in Macau for a further two more years with ThreeBond with T-Sport in 2012 and 2013. He took 15th during ’12 while improving to 4th in ’13.
2012 was a busy year for Sims as he went on to compete in the Formula 3 Euro Series and the FIA Formula 3 European Championship with ThreeBond with T-Sport as a guest. Alongside this, he joined Status Grand Prix in the European Le Mans Series and the 24 Hours of Le Mans. He didn’t manage to finish the 24 Hours however, he did finish 10th in the European Championship. He returned to the FIA Formula 3 European Championship with ThreeBond in 2013 where he raced in 12 races, taking 5 podiums and 10th in the standings. He also competed in the Macau Grand Prix with the team where he finished 4th. Alexander also joined Status Grand Prix and Carlin for the GP3 Series where he took 8th in the championship. Sims also partook in in the FIA GT Series as a guest with Boutsen Ginion Racing.
2013 marked the beginning of Sims’ time in the Blancpain Endurance Series. He joined Hexis Racing for his first year where he took 15th. During 2014, he raced with Triple 888 Racing where he placed 22nd while also joining Ecurie Ecosse as a guest in the Blancpain Endurance Pro-Am Series. He competed with the team in 2015 as a driver and placed 16th. Alexander made the move to Rowe Racing for 2016 where he raced in the Blancpain GT Series Sprint Cup and Endurance Cup. Sims took a win in the Endurance Cup to finish 4th in the standings while he finished 11th during the Sprint Cup. He returned with the team for 2017 in the Endurance cup but only managed 27th in the standings. His final year was 2018 where he joined the team for a third year in Endurance and placed 15th.
Alongside Blancpain, Sims entered the British GT Championship in 2014 with Ecurie Ecosse powered by Black Bull where he took two wins and placed third in the standings. He returned to the team and championship in 2015 where he took two more wins and finished second. Sims also partook in the FIA Formula 3 European Championship with HitechGP as a guest and in the Macau Grand Prix with Double R Racing where he finished third. Headed into 2016, Alexander joined Barwell Motorsport in the British GT Championship where he went on to finish 14th. He also returned to the FIA Formula 3 European Championship with HitechGP as a guest and to Double R Racing for the Macau Grand Prix where he placed 10th.
For 2017, Sims switched things up and made the move to BMW Team RLL where he debuted in the GTLM WeatherTech SportsCar Championship. He took three wins and 317 points to place him second in the standings. He also joined BMW as a guest in the Intercontinental GT Challenge. He returned to the WeatherTech Championship with BMW to take a further two wins and finished sixth in the standings. Sims went on to join BMW Team MTEK in the LMGTE Pro 24 Hours of Le Mans but didnt finish. To round off, he competed with BMW again in the LMGTE Pro FIA World Endurance Championship across 2018 and 2019 where he placed 19th.
Sims debuted in the FIA Formula E Championship in 2017 when he joined MC Amlin Andretti as a Development Driver. He went on to join full time when the team made the name change to BMW i Andretti Motorsport for the 2018 to 2019 season. He was joined by team mate António Félix da Costa for the year where the team went on to finish 5th with Sims in 13th.