So are you saying that the neighbour hasn't actually done anything to you - not shouted at you, insulted you, thrown things at you, had loud music, been banging stuff about, thrown things onto your property, smashed your window, screamed at your kids, etc etc - none of the things that people might have assumed you meant when you talked about a nightmare neighbour - the only thing about this person is their garden is a bloody mess and they have an old car on their property and someone else has told you they have seen rats but you haven't seen them and someone else has told you the neighbour puts rubbish outside but you haven't seen it in the 6 months you have been living there?
What has the neighbour actually done to you?
If it's just the state of the garden, then report it to environmental health if it's full of rats and you have seen them. They'll come out and have a look. They can force a clean up if they agree it needs it.
But if you just went to the neighbour and put notes through the door saying your garden is horrible, let us bring it up to scratch, can you not see why they might have said sod off? I mean, you just put notes through? Not even a hi I'm your new neighbour knock on the door? Just straight to notes?
And when you say that other neighbours have had problems - what have they reported to you specifically that is 'anti social late night behaviour'? Loud music? Shouting? Threats?
tbh, If I was bothered by the mess in my neighbour's garden, I'd just put a higher fence up and forget all about it. If I saw rats, I'd report them. If I saw rubbish being dumped, I'd report it. If the neighbours told me I'd say to them well, if you've seen that, then report it to environmental health.