Sorry to hear of the issue.
There is a solution as you are in dispute with the council. You pay council tax and yet they do not collect the waste - you have also endured missed collections so this is their fault, clearly.
You should first make a complaint to the council and you can do that here:
https://www.haringey.gov.uk/contact/council-feedback/make-complaint
They have to deal with your complaint - you can send them a letter, an email or call them - just tell them why you are unhappy and that you want to make a formal complaint and ask them to resolve this.
If you ask them to send you their final decision as soon as possible - you need that before you can then involve the Local Government Ombudsman (LGO) : www.lgo.org.uk/make-a-complaint
The LGO may be able to help but you would need to firstly exhaust the council's complaints procedure, unless you have done this already.
As it is an arbitration scheme, the Ombudsman’s decision would be binding upon the council if you accepted the decision.
I am sure this will resolve the dispute - it is also free to use and no lawyer is required either. It is set up for consumers such as yourself and is also quicker than court action.