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 a complainant is not capable of doing this themselves or do not want to do it themselves, they can charge the neighbour for the cost of doing the work provided they give the neighbour notice that if they don’t trim it, they will get someone in to do it.
They are not allowed to kill the tree or cut the roots back so far that they can 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 depending on what they are aware they are the neighbour can get the local authority to compel you to cut them down to 2 m. Please note it is down to 2 m), it does not mean that they are automatically 2 metres. Depending on the location they could be cut down to say, 4 metres.
A lot comes down to whether with them at 4/5 m, they are literally nuisance.
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I am happy to answer any specific points arising from this.
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If you still need any points clarifying, I will be happy to reply because the thread does not close. In fact, it remains open indefinitely.
I am always happy to answer any further questions you have on any new thread in which case, please start your question with, “ For FES only”.
That only applies to new threads, not this one. You have me exclusively on this one.