If I were in your shoes, I would be curious as to why a letter uses a -mildly threatening- phrase, like "legally obliged" (do tell us if that's just your interpretation of their words) whereas it doesn't take a lot of additional effort to just quote the name and paragraph of the law they are referring to.
In other words, with a poorly worded letter that I was somewhat skeptical about, I'd be ringing them up to ask for further details, just out of curiosity and 'on principle', if you get my drift.
Ultimately, I would not obstruct the work, as I might want to use the same legal position in my favour, perhaps with the same neighbour being on the receiving end instead.