Honestly, where do people get this utter drivel from? Stop making shit up.
Maintenance of the tree is the responsibility of the landowner (yes, even if the tree was there when you moved in). Therefore, the landowner pays. If the landowner refuses to pay, the neighbour can under their common law rights cut back to the boundary, UNLESS the tree has a TPO on it, in which case the landowner has to apply for permission from the local authority.
If the landowner refuses to pay, then the neighbour will have to stump up (see what I did there). HOWEVER, the landowner has a responsibility to ensure that their tree is safe. See this advice:
The tree owner also has a duty under the Occupiers Liability Acts
to take reasonable steps to ensure visitors or trespassers on their land are safe. In practice this means that if a tree fails and causes damage to a person or property then the tree owner may be liable.
Ideally, unless you’re a selfish cunt or a twatty neighbour, if a tree is overhanging your garden enough to cause you problems, then you should go and speak to the tree owner, explain the impact it’s having and ask that they cut it back. If you think the owner won’t be amenable, then offering to contribute towards the cost can help oil the wheels but you are not required to be law.
Though because this is Mumsnet, people will seethe silently then claim they have anxiety, stress and don’t do confrontation, so of course it will escalate.