Bulls chairman David Keyte stated in March that he felt the club's five-man coaching staff was too many for the long-term and action has now been taken following the club's relegation from the Football League.
Hoult's contract has been terminated with immediate effect, with a settlement now due, while Sneekes' 12-month deal will not be extended.
First-team coach and former boss Jamie Pitman is set to remain along with director of football Gary Peters, while talks continue with manager Richard O'Kelly over his future at Edgar Street.
Keyte told the club's official website: "On behalf of the board of directors I would like to thank Russell and Richard for their work at the football club.
"They are both great characters to have had around the club, but we must cut our cloth now to get ready for life in the Blue Square Bet Premier."
