Nelson played in the Wisconsin Interscholastic Athletic Association for the Wisconsin Rapids Lincoln High School Red Raiders at the SWCRC (South Wood County Recreation Center) in Wisconsin Rapids, Wisconsin US.
Nelson played in the North American Hockey League with the Alaska Avalanche and Johnstown Tomahawks. He was named the NAHL Defenseman of the Month for November 2012, and he played in the 2013 NAHL Top Prospects Tournament.
Nelson joined the NCAA‘s Minnesota State Mavericks in 2013 and played three seasons at the school. He was named to the All-WCHA Second Team in the 2014–15 season. In the 2015–16 season, he was named the conference’s Defensive Player of the Year while being named to the All-WCHA First Team.
Nelson played alongside his brother, Josh, for the 2013–14 season.
Nelson was signed to an entry-level professional contract by the NHL’s Buffalo Sabres on March 22, 2016. He made his NHL debut with the team on March 26, tallying his first NHL point on an assist in the first period of a game against the Winnipeg Jets.
On June 29, 2016, Nelson signed a two-year contract extension with the Sabres. He recorded his first NHL goal on February 11, 2018, in a 5–4 loss to the Colorado Avalanche. On May 2, 2018, Nelson re-signed with the Sabres.
As a free agent, Nelson initially returned for a sixth season within the Sabres organization by agreeing to a one-year, two-way contract extension on October 22, 2020. However, Nelson later opted out of participating of the delayed 2020–21 season due to the ongoing COVID-19 pandemic.
On January 15, 2021, Nelson announced through Instagram that he was retiring from hockey after five professional seasons.