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.
with regard to the right to light, it needs to have blocked out 50% of your light before it becomes actionable. I can tell you now, that is unlawful what of light.
The problem for you is that if the neighbour simply ignores you, you can’t beat them with a stick until they capitulate and your only remedy is to take them to court. The court fee alone is GBP208 although if you are on benefits you can actually get fee remission. There is no magic way of getting the neighbour to capitulate and sort the trees if he simply doesn’t want to do it or refuses, regardless of the reason.
Can I clarify anything else for you?
I am happy to answer any specific points arising from this.
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