Former Leeds United and Rotherham United manager and Barnsley legend leaves National League club
Thompson was put in caretaker charge following the sacking of David Unsworth last month, with the Lancastrian soon bringing in Redfearn to assist him.
Thompson had expressed a desire to land the job on a full-time basis at Latics, who were unbeaten in six matches in charge under him and Redfearn, winning three matches and drawing three.
But a club statement announced the pair's departure on Thursday morning.
"We would like to place on record our thanks to Steve and Neil for their work in taking charge of the first team during recent weeks.
"Academy manager Paul Murray will oversee training ahead of Saturday’s Emirates FA Cup tie against Altrincham.
"An announcement in relation to the managerial position will be made in due course."