Swift was born in Rotherham, South Yorkshire, he began cycling with Mossley Cycle Racing Team aged 12. In 2003, he came second to fellow Olympian Steven Burke in the under 16 national scratch race championship. As a junior rider, Swift competed at the Junior Commonwealth Games, taking bronze in the points race, he won two national titles, the junior points race championship in 2004 followed by the junior scratch race in 2005. He also competed in the senior scratch race in 2005, demonstrating his ability by finishing third to take the bronze medal. Swift began competing internationally, winning several madison events. In 2012 he won the World Championship Men's Scratch Race, becoming Great Britain's first ever World Scratch Champion.


He made his professional road debut in August 2007 joining Barloworld as a trainee during which time he won the King of the Mountains title in the Tour of Britain.

During 2008 he raced for the British Cycling development team and won his first UCI ranked race. He represented Great Britain in the 2008 Olympic Games road race and at the UCI Road World Championships where he finished fourth in the under 23 race.

After the championships he signed a two-year professional contract with the new Russian Team Katusha.

In 2009 he was selected to ride the Giro d'Italia, taking a third-place finish in stage 2. He took his first professional win on the seventh stage of the Tour of Britain, leading Katusha teammate Filippo Pozzatoin a one-two finish. Swift spoke with in 2009, describing himself as "an allrounder", who can do well in the Classics and win stages. "My weakness is in the time trial. I don't like doing them and I'm not really that good at them".

Swift joined the newly formed British-based Team Sky in January 2010. His transfer to Sky from Team Katusha part way through his contract was instrumental in the Union Cycliste Internationale considering new regulations for the transfer of riders between teams. During his first season with Sky Swift's most significant result came at the Tour de Picardie, at which he won one stage, the points classification and best young rider.


After seven seasons with Sky, in September 2016 Swift announced that he would join UAE Abu Dhabi on a two-year deal from 2017 in order to seek more opportunities to ride for himself.

In October 2018, Swift rejoined Team Sky for the 2019 season. In June 2019, Swift won the British National Road Race Championships in Norwich, beating team leader Ian Stannard. In September 2019, Swift signed a new contract to remain with Team Ineos until the end of the 2021 season. Swift said he had 'found a renewed enthusiasm and motivation for the sport'.