I'm not too sure that mentioning an overhanging branch or roots to the council will lead to them getting involved at all. That's purely a personal thing between two neighbours.
If roots from the tree were damaging the pavement out front then it is likely that the local council would step in, but not otherwise.
"Financial and physical assistance have been repeatedly offered but he steadfastly refuses."
If that is the case then there is little that you can do. What you can do is to cut back any of his plants that come onto your property.
So, for example, with the oak tree (and any other trees and hedges) you can cut back any overhanging branches and roots back to the property line.
You don't need his permission to do this - if it's overhanging your property or the roots are going under your ground then it's a "nuisance" and you can remove them back to your property line. You'll have to do the work yourself or pay to get it done.
What you cannot do without his permission is to go onto his land and start chopping things down.
As others have said, with the issue of rats then, yes, contact the Environmental Health dept of your local council. however, it is likely that they will only investigate where there is good evidence of an active infestation. You will need to supply plenty of evidence about this.
One way may be to contact a reputable pest control company, they will likely be able to find the evidence of rats if they are coming on to your property.