This is the same legal question from the other side of the fence, and the answer is symmetrical: your neighbour has a limited right, and it stops at the boundary.
What they are entitled to do
Your neighbour may cut back branches and roots that cross onto their land, as far as the boundary line. They do not need to ask you, and they do not need to show the overhang is causing a problem.
That right exists regardless of who planted the tree or how long it has been there.
What they are not entitled to do
They may not cut beyond the boundary. Reaching over to cut back to the trunk, or taking a limb that does not overhang, is damage to your property.
They may not come onto your land to do the work, or place a ladder in your garden, without your permission.
They may not do anything at all if the tree has a Tree Preservation Order, without written council consent. A TPO prohibits lopping and topping, and it binds your neighbour exactly as it binds you. The same applies to the six weeks' notice required for a tree in a conservation area.
And the cuttings remain your property. They should be offered back to you rather than disposed of or thrown over the fence.
If they have already gone too far
Photograph everything, from several angles, and note the date. If the cutting has been severe, get an arboriculturalist's assessment of the damage and the effect on the tree's stability and long-term health. That report is what turns a dispute into a claim.
If the tree was protected, tell the council's tree officer. Breaching a TPO is a criminal offence and enforcement is the council's, not yours. Penalties on conviction on indictment are unlimited.
Where the tree was not protected, your remedy is civil: a claim for damage to your property. Speak to a solicitor before spending money on anything else, because the cost of litigation over a tree frequently exceeds the value of the tree.
Preventing it
A conversation is worth more than a solicitor's letter. Most neighbour tree disputes escalate because nobody raised it early.
If a neighbour has complained about overhang, consider having the tree pruned properly yourself. A balanced reduction by a competent arborist costs less than the eventual dispute, keeps the tree healthy, and removes their motive for taking a saw to one side of it.
If you know the tree is protected, tell them, in writing, before anything happens. They may genuinely not know, and a homeowner who cuts a protected tree in ignorance is still committing an offence.
Light and roots
There is no general right to light for a garden, so a neighbour usually cannot compel you to reduce a tree because it shades their garden. Evergreen hedges over 2m are the exception, dealt with under the high hedges provisions.
Roots follow the same rule as branches. A neighbour may cut roots crossing the boundary, though cutting substantial roots can destabilise a tree, and if it later fails the person who cut them may find themselves liable.