Sorry to hear of the problem with BT.
There is a way to sort this out. You can apply to the telecommunications Ombudsman - which is free to use and is quicker than court action.
It is an arbitration scheme meaning the Ombudsman's decision is legally binding if you accept it.
You do have to make a formal complaint to BT first before you can avail of the Ombudsman though. The Ombudsman will not help until you have BTe's final decision which you now need to ask for.
You can allege breach of contract against BT (you have consumer rights here which have been infringed in my view).
You can make the complaint and ask them for the final response via this link: https://www.bt.com/help/contact-bt/complaints
Once you have the deadlock letter the next step would be to escalate this to the Communications Ombudsman - they will investigate and liaise with BT which would hopefully result in a resolution. They can also order the company to make a financial award if they have acted poorly.
The service is covered by the Consumer Rights Act 2015 so you have a right to expect a reasonable service - if they do not provide that then the regulator will step in but it also means you can sue for any damages or losses incurred as a result o BT's conduct.
You can escalate to the Ombudsman here: www.ombudsman-services.org/sectors/communications
Or by email:***@******.***
If you have lost any money then you could sue BT but the court may consider that to be premature if you do not go through the Ombudsman first of all.
You can sue for your losses via the http://www.moneyclaim.gov.uk website which is easy to use and no lawyer would be required.