Williams is the new head coach of the county.
Ian Burchnall left Meadow Lane in May to become the head of Forest Green Rovers.
He left his job in south Wales in February for personal reasons.
Williams is fully aware of the pressure and expectation that comes with this role, according to a statement from County.
The club were demoted from the English Football League for the first time in their history in 2019.
In their three seasons in non-league football, the Magpies have reached the National League play-offs in two of them.
Ryan Harley is the assistant to Williams.
They have the right management skills and experience of the National League, which we believe will help us achieve our goal of promotion and getting back to where we feel we belong.
At the end of his first full season in charge of the club, Williams ended his tenure as manager.
He was given a five-year contract after impressing as interim boss.
He was the head coach of the Bristol City Under-23s for two years before taking the job as Russell Martin's assistant at the Dons.
He left in six months after following Martin toSwansea.