People make all sorts of empty legal threats and demands – it’s a form of bullying. As such, there is only one appropriate response: bring it on!
There is no statute that says switching off a device constitutes criminal damage. Personally, these people have been so unhelpful and selfish, I’d be taking the whole thing down.
Meanwhile, have a look at this pamphlet on neighbourhood noise. Your neighbours should not be operating the air-con between 8pm and 7 am (weekdays) if it can be heard in your flat, and could be fined for doing so.