Gosh, that must be scary for you.
There are two avenues you can pursue with your local council. Firstly the Planning Department: look for ‘planning enforcement’ who are the people who check the legality of the structure’s right to be built.
The other department are Building Control, who it sounds like you have contacted. They are the people who check building work for structural integrity etc. The two departments don’t necessarily communicate, so it’s worth speaking to both.
The distance from the boundary is something you might want to enquire about, as well as height of the structure.
It might be possible to get advice/legal help from your building insurance. If you rent the property, let your landlord know, as they might have an avenue to help.
Citizen’s Advice might be another possibility. It’s really good that you have photographic evidence, we can see quite clearly your problem.
Always try to complain on breaches of planning law as it holds more weight. All the best.