Walker was born in Cardiff, Wales, but grew up in Australia and became the first Australian player selected by an NHL team when the Washington Capitals chose him in the third round (No. 89) of the 2014 NHL Draft.

Walker moved to the Czech Republic in 2007 after dominating local competition in Australia. After six years, he signed with Youngstown of the United States Hockey League in January 2013 and finished with 27 points (seven goals, 20 assists) in 29 games. He played the next four seasons for Hershey in the American Hockey League before making the Capitals at training camp in 2017.

With his parents and brother in the crowd, Walker scored in his NHL debut Oct. 7, 2017. He was waived by the Capitals on Nov. 30, and claimed by the Edmonton Oilers the next day. But the Oilers waived him on Dec. 19, and he was reclaimed by the Capitals.

Walker made his Stanley Cup Playoff debut in Game 6 of the Eastern Conference Second Round against the Pittsburgh Penguins and had an assist, becoming the first Australian to play and have a point in an NHL playoff game.

After six seasons with the Capitals organization, Walker signed a one-year, two-way contract with the St. Louis Blues on July 1, 2019. He signed a two-year, $1.775 million contract ($887,500 average annual value) with the Blues on Sept. 17, 2025, which was to begin with the 2026-27 season. His 16 points (eight goals, eight assists) in 73 regular-season games were an NHL career high and his three goals in seven playoff games tied Jordan Kyrou and Pavel Buchnevich for the team lead.