If the disrepair of a householder’s roof presents a danger to members of the public then the responsible party should be forced to take action.
The first course of action in disputes with neighbouring householders is always negotiation. The neighbour should always be given the chance to resolve the problem themselves before the issue is taken up with an official body. You say that the householder is aware of the problem but they may not be aware of how serious a danger it presents – a falling roof tile is no leaf, it can cause serious injury or damage.
The problem of the poorly maintained roof, its potential risk to safety and damage, and the householder’s legal responsibilities needs to be outlined to the neighbour. Face-to-face communication can often be the quickest and most straightforward approach. However, if you are uncomfortable broaching the issue with the neighbour directly then it is perfectly fine to address it in the form of a letter.
The advantage to this method is that it will allow you convey the problem in a clear headed and rational manner, relieved of the heightened emotions of a doorstep confrontation. A letter will also provide you with proof that you have attempted to negotiate with the neighbour if the matter remains unresolved and you then decide to pursue it at a higher level, so remember to keep a copy.
Having been informed of the danger of their loose slates, the householder would then be liable if the slates were later to fall and cause injury or damage. This would be classed as negligence in a court of law.
If your efforts prove ineffective and the neighbour is unwilling to cooperate then your next step would be to contact your local council. The complaint would fall under the jurisdiction of its Building Control team, or Environmental Health Department. If the loose slates, as you say, present a public danger then you have a very valid complaint that should prompt immediate action.
In the unlikely event that this course of action fails to achieve a resolution then the next step would be to bring a civil case. This is a complex issue that would require you to seek legal advice.