Hi everyone. I don't know if any of you have watched a programme on tv about the council's housing officers, but similar problems have been highlighted on this programme. In one instance, after years of requests and negotiations, the owner was given the option of sorting out the property (which had been left unoccupied for years) or handing it over to the council. In other cases the housing officer attended and made the owner not only make the property safe but also to clean up the property. It would appear that they do have some powers to make owners at least make properties safe and not encroach on a neighbours property. As you say, rats are a risk and very likely to be 'living' in the neighbours garden. You might also try the Environmental Health department as it can't be a very healthy environment to be next door to.
As for the ivy. The best way is to cut the growing stems/branches as near to the ground as possible, which will result in the ivy dying and making it easier to remove. Copper nails pushed into the base of thick stems will also work in the same way I'm told, although a bit slower, and dare I say, make it less obvious that it has been touched!! This way there is no need for weedkiller on the ivy at least.
Hope this helps and good luck.