The question is, how does one "nip it in the bud". Whilst in a perfect world the police will turn up, have a quiet chat about their behaviour and everything will be good again, this is not a perfect world. If it was just a one of rant, I would let it lie and it may well never rear its ugly head again. Perhaps nathan could have a quiet friendly chat with the neighbours?
However, if her behaviour continues, yes do contact the police. The difficulty as I foresee it is if you start making complaints to the police and attempt to have this woman charged, it is not going to help matters long term. Mum will be left living next to a neighbour who will be persistently hostile.
A similar thing happened to me a few years ago when a neighbour put an extremely offensive note through my door. Although my initial reaction was to make a complaint to the police I decided to let matters lie and instead had a very quiet and polite word with them. We then became reasonably good neighbours to each other and would have a chat in passing. OK, we were never good friends, but at least we managed to find a way to live next door to each other. Had I made a formal complaint, I am not sure that would have been the outcome.
In my experience, neighbour disputes have the propensity to be some of the most vitriolic, vengeful and stressful types of case. Recourse to the police and courts is one option, but not always the best option.