Yes, they can be prosecuted, RP. Under the new Animal Welfare Act, the dog owner would be liable if:
(a) an act of his, or a failure of his to act, causes an animal to suffer,
(b) he knew, or ought reasonably to have known, that the act, or failure to act, would have that effect or be likely to do so,
(c) the animal is a protected animal, and
(d) the suffering is unnecessary.
Not only could he reasonably be expected to know that his dog could kill a cat, he has been TOLD of it, he KNOWS his dog escapes and it is KNOWN that the dog hates cats. Therefore he is in deep shit should his dog kill your cats BUT... you don't want it to come to that.
If he is allowing his dog to continue to escape he isn't nice, no matter how friendly he comes across. He's putting your cats, your neighbours cats AND his dog at risk of harm. You've two choices as far as I can see it - either print out and hand him a copy of the Animal Welfare Act, together with a copy of a letter you have written telling him that his dog continues to access your garden despite you repeatedly asking him to contain the pooch and that he has acknowledged both that you have cats and that the dog is aggressive towards cats. Point out that the dog is trespassing as he has a duty in law to keep the dog under control and that you are keeping a copy of the letter and will be bringing charges under the Act if your cats are harmed by his dog. Add that you are sending a copy of the letter to the dog warden - and make sure you do.
Otherwise, get out there fast and put a fence up around ALL of your garden - even just a chainlink one from B&Q and meantime call the dog warden and ask him/her to visit you to view and advise and then to visit the neighbour to issue advice and/or a warning.
Personally I'd do both of the above on the grounds of a belt and braces approach as AFAIAC there is no excuse for allowing your dog to roam into other's gardens. The owner should take him into the garden on lead if the garden isn't secure.
It ain't rocket science, is it?!! :o Not to do so is pure bloody idleness which could cost a creature his life.
HTH
Val (with love from Esme, who is currently asleep on top on the Sky TV box, under the TV cabinet!).