The 41-year-old was previously head coach at Swindon Town and most recently worked alongside Russell Martin at Swansea City.
County have been working hard to find the man to lead them forward after Burchnall departed for League One Forest Green Rovers.
He will be assisted at Meadow Lane by Ryan Harley.
A club statement read: “We’re delighted that Luke and Ryan have joined us as our new head coach and assistant head coach.
“Between them they have the right management skills and experience of all EFL divisions, as well as the National League, which we believe will help us achieve our goal of promotion and getting back to where we feel we belong.
“Luke's values and playing principles are aligned with our own and we feel the continuity and consistency this brings will be a huge factor in ensuring a smooth transition to his leadership.
“We have been very impressed not only with Luke’s general character, but also his excitement about this opportunity and his fierce determination to succeed.
“He fully understands the pressure and expectation that comes with this role and we can see he’s ready to take on the challenge.
“Luke has already met with our recruitment team to discuss our current squad and ongoing transfer targets and we look forward to working closely with him to build a team which will give us the best possible chance of earning the promotion we all crave."