It would be classed as a 'near miss' in industry and then an investigation would take place to try and prevent it happening again.
Some remedies may be, a/ making sure the work area is cordoned off securely to prevent anyone entering the danger zone. b/ Possibility of wrist straps on hand held tools to prevent them falling. c/Safety netting around the edge of the structure to catch anything falling.
This is not a reportable incident because no -one was injured/killed in this instance. However, the builder should investigate how/why it happened and try to mitigate it happening again. As he freely admitted it was his responsibility, and was willing to make reparation at his cost, the HSE wouldn't even bother investigating. If his offer has been refused by the neighbour, then he could give the neighbour a written report of his investigation and tell them they are free to contact HSE themselves to discuss the matter. Anyone can report an incident to HSE, it does not have to be the builder, the site owner who contracted the builder or any of the employees. If a member of the public had seen the incident, they could report it if they wanted to. Throw it at the neighbour and tell him/her to report it and you/the builder will act on their recommendation. (They won't bother giving one).