Sorry to hear of the issue.
No, it's only the council who deals with housing disrepair issues. But you should use a law firm (we are not a law firm, we are an internet forum) to sue for specific performance (ie. force the landlord to carry out the repairs) and also to claim compensation.
The council's housing department should be notified of the issues you have - I note you say you are speaking to them on 7th July which is fine. They will want to inspect the property. They may serve an improvement notice on the landlord to force them to carry out the works.
My view however is you use a law firm to pursue a disrepair claim. You may well be offered no win no fee terms - meaning you pay nothing up front. The lawyer pays the court costs and you are also protected from having to pay legal costs if you lose.
With a disrepair claim you can also claim compensation for loss of enjoyment.
I would not recommend you withhold rent because this then gives the landlord a reason to evict you. Far better to use a lawyer to sue the landlord and claim some rent back, with compensation on top.
As for the law, in UK law, landlords must ensure the property is in good repair under Section 11 to 16 of the Landlord and Tenant Act 1985. Further, a recent law came in to force to provide more protection to tenants, which is the Homes (Fitness for Human Habitation) Act 2018 is now in force - the house or flat needs to be healthy, safe and fit for human habitation (free from things that could cause serious harm).
If they do not, they face prosecution by the local council and can be fined. It would also be a breach of contract on their part. The only situations where a tenancy can be ended are, you finish the tenancy term or there is a break clause meaning you can end the tenancy part way through. Or the landlord agrees to you leaving early - usually if another tenant can be found. You could terminate but you would need evidence of repeated requests to the landlord to fix and they do nothing. If you terminate then they could sue and you may have to defend the claim in a civil court.
The Local Authority Environmental Health Officers have powers under the Housing Act 2004. They have a duty to ensure that properties in their area are in a habitable condition, and will serve improvement notices on landlords of properties which are assessed, under the new Housing Health and Safety Rating System as having ‘category 1 hazards’.
If those notices are ignored, the council has the option to prosecure the landlord in the magistrates court - which can lead to a criminal conviction.
Similar powers are available to Local Authorities under the Environmental Protection Act 1990 if the property is considered a threat to public health.
If you have to sue the landlord, you may want to use a no win no fee lawyer - there is nothing to pay up front and you are also protected from legal costs if you lose.
Here are a few choices for you if you would like to use a lawyer (if you ask for the housing team):
1. Brown Turner Ross, tel: 01704 542002 (housing department)
2. Stephensons Solicitors, tel: 0161(###) ###-####/p>
3. Duncan Lewis, tel: 0333(###) ###-####/p>