Carruthers and Mark Clifford parted company with the EVO-STIK League Premier Division side on Wednesday.
Hearn will be supported by Dom Roma as interim coach and Graham Rodger, who joins from AFC Mansfield as goalkeeping coach.
Vice chairman Tom Brookbanks takes on a new role as Director of Football.
On the playing front, Basford have signed centre back Brad Gascoyne from Barwell.
Chairman Chris Munroe said: "I am delighted with these appointments and signing.
"Liam and Dom are both highly experienced Football League and high National League level players.
"Liam has already shown excellent potential managing our Educational Academy team, and Dom managed Gainsborough Trinity in National League North.
"The fit is good and they will command the respect of our excellent playing staff.
"Myself and Tom will work closely together, tapping into his great experience and knowledge to formulate the future development and direction of this great football Club."
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