Prosecutors said the firm continued to trade despite being insolvent and unable to meet its obligations.
Elkin was additionally convicted of making and using a false instrument by displaying a forged certificate from the National Association of Funeral Directors, they added.
Sharon Kirton, who was a client of the firm, said she was worried about how her mother’s body had been treated.
Speaking outside the court, she said: “It will never take away the pain we’ve got.
“My mum said: ‘Make sure I’m looked after. I feel like I’ve let her down.'”
Rachel Robertson from the Crown Prosecution Service (CPS) said the pair, of Nobes Avenue, “showed a grave disregard for the dignity owed to the deceased in their care and the trust placed in them by grieving families”.
The CPS said the mortuary was left unrefrigerated between June 2022 and December 2023 and many of the bodies were left there for more than 30 days.
Assistant Chief Constable Paul Bartolomeo said: “Our officers turned over every stone to bring Elkin & Bell to justice using legislation that is hundreds of years old.
“Sadly we are aware of other similar cases across the country
“We need new legislation rather than relying on common law.  We also need better regulation.
“This can help ensure that all funeral directors act, as the majority do, with professionalism and compassion.”