Contact your local council planning department, they should come out and offer advice and a solution they should also be able to tell you who's responsibility it is.
Or as i've said in another post - it may be petty - but if the neighbour is unapproachable or isn't bothered. I would install edging stones or a wall then the water build up would be their problem - you'd soon see the problem sorted then
Check your deeds for covenants. I live "downslope" from my neighbour who has "right of drainage". Equally I have right of drainage over the next neighbour down the hill.
If that's the case you can do nothing about it. Blocking it off isn't an option because you deny the "right" if it exists.
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