Chairman Alwin Thompson said: "I am very pleased we're sitting here with Paul because I think it's a great appointment to take this club forward."
Simpson will be assisted by Peter Ward and Alan Lord, who were coaches under former manager Jim Gannon.
"I'm delighted to be coming into a club which is starting a new era and hopefully what will be a new era for the football club," Simpson told BBC Radio Manchester.
"That's what attracted me - the challenge. I've had a couple of months out of work and this is a great challenge.
"This is a club that's been on its knees and the new chairman and directors have come and picked it up and they're setting off on a journey I'm delighted to be part of. I want to make it a success."
Ex-boss Ablett lasted one season at County with seven wins from 51 league and cup games as they finished bottom of League One.
Former Rochdale, Carlisle and Preston boss Simpson was sacked by Shrewsbury in April after failing to lead them to the League Two play-offs.
The 43-year-old took Shrewsbury to the Wembley final in 2009 but paid the price for failing to finish in the top seven last season and lost his job after just over two years in charge.
The guy's worth another opportunity. But he did get things so wrong at Rochdale, Preston & Salop
Jenny
He started his managerial career at County's neighbours Rochdale in 2002 but his most successful spell came at Carlisle, whom he led to back-to-back promotions from the Conference to League One in 2005 and 2006.
He left Carlisle to join Preston just weeks after leading them to the third tier but was sacked after less than 18 months, with North End sitting near the Championship relegation zone.
The former Oxford midfielder then took charge of Shrewsbury towards the end of 2007/08 and led them to the play-off final in his first full season in charge, where they lost to Gillingham.
And he was sacked earlier this year with the Shrews five points off the play-offs with two games left.
Source: BBC Sport
Source: BBC Sport