He replaced Neal Ardley, who left following Tuesday night’s draw at Yeovil Town with the Magpies sitting in sixth in the National League.
Burchnall, 38, arrives after working in Scandinavian football for eight years.
He teamed up with now-Brighton manager Graham Potter at Ostersund FK and Notts’ new owners feel it is time he had a go at English football.
Burchnall could take the club to Wembley in his first match - he will be in charge for Saturday’s FA Trophy semi-final with AFC Hornchurch.
Owners Chris and Alex Reedtz said in a statement: “We are tremendously excited to welcome Ian to the role of head coach.
“As part of our ongoing extensive research, we compile detailed profiles of many different coaches and Ian stands out as an outstanding prospect.
“Despite his young age, he has tremendous experience of working in top-flight football and has been linked with several international and club roles since his departure from Ostersund. We feel very lucky to have him and have every confidence that he is the right man to complement our model.
“It’s our hope that Ian will be able to continue our push for promotion this season but we understand it will take him time to fully implement our philosophy. He is a long-term appointment for this club and his performance in the role will be judged on that basis.
“We think supporters will share our appreciation of Ian’s passion for coaching and his willingness to take risks and meet difficult challenges head-on, as he did during his time in Scandinavia, and we’re very excited to see him lead the team for the first time in Saturday’s Buildbase FA Trophy semi-final against AFC Hornchurch.”