Many thanks for your patience. The starting point is that the tradesman would be responsible for this because it was their negligence which caused the damage. They should cover themselves with liability insurance but if no such insurance exists to cover the damage, they can be held personally responsible for this.
In terms of Checkatrade they will have limited liability here – they may have checked that the tradesman has insurance, which would have been true, but the issue here is not that they did not have insurance but that they had no specific cover for the type of work which was carried out. So their check may have been a basic one to see that some form of liability cover is in place, but without having to check the exact specifics of what that covers. So it is unlikely that they can really be held responsible here.
At this stage the main liability lies with the tradesman and if the neighbours can show that your actions were negligent then they could potentially pursue you too, at least for part of the costs. What is negligent is something only a court can decide but it would be considering things like whether someone in your position would have been reasonably expected to check the details of the level of cover the tradesman had, although I would argue that it is the tradesman’s responsibility to ensure they had adequate cover based on what work they were doing so the main liability will lie with them.
Finally I am not sure why they are pursuing you for such a large figure – was that much damage actually caused? They cannot penalise you for what happened, they can only pursue the guilty party for the actual damage and losses incurred so they will have to justify that £250k of damages have been caused to be able to go for that amount.
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