This is a very common question. A person cannot make an adjoining property owner cut the trees per se but please see my later comments about trespass and nuisance.
Consent is needed to trim deciduous trees which are protected by a Tree Preservation Order and a licence may be needed to fell trees which are not protected by a TPO.
Evergreens cannot usually have a TPO.
Regarding evergreens, a complaint can be made to the council and they will deal with them under the nuisance tree legislation but only with regard to the height. They want £300 fee to start the process and if they find that the trees are a nuisance, they can compel a tree owner to cut them down to 2 m high.
With regard to branches and roots growing boundary,(evergreen or deciduous) these are nuisance and trespass. The overhanging pieces can be chopped off as can roots growing underground but they do not belong to anyone other than the tree owner, so the pieces should be given back although if they are unceremoniously dumped over their hedge without warning, it is not good for already fraught neighbour relations.
If you are not capable of doing this yourself or do not want to do it yourself, you can charge the neighbour for the cost of doing the work provided you give the neighbour noticed that if they don’t trim it, you will have to get someone in to do it.
You are not allowed to kill the tree or cut the routes back so far that you destabilise it.
The following are links will give you some reading with regard to high hedges and nuisance trees. Don't worry about where the sites are geographically because the rules apply nationwide.
So what you have been told by the council is not correct. They can do something about it. I would suggest that you now put your complaint in writing to the council and tell them that if they don’t know anything about this you will make a complaint to the Local Government Ombudsman.
You can of course bring a claim against the neighbour yourself in common law private nuisance but to be frank, there is absolutely no reason why you should have to go to that trouble and risk.
You may find that a strongly worded letter from a solicitor threatening a court application and an application for costs, may focus the mind without actually the need to get to court.
Check your house insurance to see if you have legal expenses cover that would pay for the legal cost of taking this to court.
Having said all this, the council should deal with this for you. You should not have the trouble of having to go to court or to be frank, even threaten it.
Can I clarify anything else for you?
I am happy to answer any specific points arising from this.
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