You have no contract or relationship with the firm that did the work, and they may not have any duty to you (their duty is to their customer). Further, you can't rectify any part of the roof on the neighbour's side of the boundary, unless they explicitly give permission - and if they say "Yeah you can fix it" be careful that that is not agreement for you to fix it at your cost.
What you would do is invite the neighbour to rectify his roof. If he declines, then you have options
- offer to do it and agree costs,
- apply to the county court for an injunction forcing the neighbour to rectify it
- carry out works yourself and sue the neighbour in the county court for your losses
- get the council to take action under their powers - they could force the neighbour to rectify the roof within 21 days, or face action in the magistrates court. This costs you nothing.