Hello, I am Keith, one of the experts on Just Answer, and happy to help you with your question.
Warn them that unless your organisation is paid you will be taking an action in the County Court. Then find a local, trusted solicitor to take the action on your behalf. That should concentrate their minds somewhat.
Also I suggest that you tip off your local media outlets as to the Sheffield City Council's current antics and how it is adversely affecting your disabled customers.
I can see how this is affecting Corporation Tax bills and cresting a cash flow problem. Once again put pressure on the Council for payment.
I do hope that I have shown you a way forward with your problem.