[2] Riley's brother, Brendon Nash, played for the Montreal Canadiens and the Hartford Wolf Pack in the American Hockey League (AHL).

Instead, Nash went on to play college ice hockey at Cornell University of the NCAA's ECAC conference, where he was a linemate of future NHLer Colin Greening.

Nearly one month later, on July 19, Nash signed a three-year, entry-level contract with the Hurricanes, forgoing his final year of eligibility at Cornell.

[6] Following the signing, Nash joined the Hurricanes training camp[7] but was reassigned to their American Hockey League (AHL) affiliate, the Charlotte Checkers, to begin the 2010–11 season.

[22] After six seasons with the Hurricanes organization, Nash left as a free agent to sign a two-year contract with the Boston Bruins on July 1, 2016.

[23] Following the signing, Nash joined the Bruins for their 2016 training camp[24] and played on their fourth line alongside Dominic Moore and Danton Heinen for their opening night.

[1] At the start of the 2017–18 season, Nash replaced an injured Patrice Bergeron on the Bruins top line alongside Brad Marchand and David Pastrňák.

[32] While the Bruins qualified for the 2018 Stanley Cup playoffs, Nash missed Boston's final five games after a misfire shot by teammate Torey Krug resulted in more than 40 stitches to close a cut near his ear.

[39] While on injured reserve, Nash was traded by the Blue Jackets to the Toronto Maple Leafs for a conditional seventh-round pick in the 2022 NHL Entry Draft on April 9, 2021.

[40] Remaining on the sidelines for the regular season, Nash returned to play in the postseason, making his Maple Leafs debut, going scoreless in two contests of a first-round defeat to the Montreal Canadiens.

As a free agent from the Maple Leafs, on July 31, 2021, Nash remained in Canada by agreeing to a one-year, $750,000 contract with the Winnipeg Jets.

On December 7, Nash's tenure with the Jets ended as he was claimed off waivers by the two-time defending Stanley Cup champions, the Tampa Bay Lightning.

On March 21, 2022, he was traded back to the Lightning in exchange for future considerations as added forward depth insurance and he was immediately re-assigned to AHL affiliate, the Syracuse Crunch.

[46] He enjoyed an offensive resurgence in the AHL with the Charlotte Checkers in the 2022–23 season, lead the team in scoring with 24 goals, 35 assists for 59 points through 66 games.