Collections stopped on Tuesday and won’t begin again until the service resumes.
Residents are asked to use the borough’s household waste recycling centres to dispose of their green waste.
Coun Chris Lamb, cabinet spokesperson for environment and transport, said: “Like many other councils who are taking this step, we have increasing numbers of staff off work with positive Covid-19 cases and isolations.
“We’re moving staff from non-essential services to focus emptying general waste and recycling bins.
“This pandemic is far from over, and we’re in a situation where staff absences are the highest since the start of the pandemic.
“Our staff must follow government rules and self-isolate to protect themselves, their families and our residents.
“We know that this isn’t ideal and will restart the green bin collections as soon as possible.”