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.
To get the height of the trees reduced if the trees are deciduous, you are going to need some evidence from Ann Arbor culture wrist or tree surgeon which confirms that although the trees may be healthy (we are assuming they are) they need considerable maintenance and potentially felling as they could be dangerous in a gale.
However if even the threat of court proceedings for a civil injunction to get the trees cut down fails to have an effect on the neighbour, you cannot beat her with a stick until such time as she capitulates and allows the trees to be cut down or orders them to be cut down and your only remedy is then to take them to court.
If she has no mental capacity, she doesn’t escape liability because she will still have assets that could be used to pay for any litigation or damage.
What you may find is cheaper is to offer to pay all or part of the cost of reducing the height of the trees although if she is a naturalist/environmental enthusiast, she may think that the bigger the trees, the better. That is of course until such time as it falls down, causes damage and she loses everything she has to pay for the cost of the damage.
To avoid her claiming that the trees could not be reasonably foreseen to fall down, you need that specialist report to confirm that this is reasonably foreseeable and to put her on notice of the same.
Apart from any evergreen trees which the local authority can deal with, the deciduous trees are indeed a civil issue.
Check your house insurance to see if you have legal expenses cover that would cover the cost of the litigation.
I know this doesn’t give you a magic answer to the problem and the reason is quite simply that there is not one. The question is surprisingly common.
Can I clarify anything else for you?
I’m happy to answer any specific points arising from this.
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