Branches crossing a boundary are a nuisance in the legal sense, and English common law gives you a right to abate that nuisance yourself. The right is real but narrower than most people assume.
What you may do
You may cut back growth that overhangs your property, to the boundary line.
You do not need your neighbour's permission to do this, and you do not need to prove the branches are causing damage. The right applies to roots crossing the boundary as well as branches.
The three limits
Cut to the boundary, not beyond. You may remove what is on your side. Reaching over and cutting back to the trunk, or taking more than overhangs, is damage to your neighbour's property and they can sue for it.
Do not enter their land. You have no right to go onto your neighbour's property to do the work, or to lean a ladder against their fence, without their permission. Trespass is a separate wrong from the tree question.
The cuttings are theirs. The branches remain your neighbour's property even after you have cut them. You should offer them back. Throwing them over the fence is fly-tipping onto their land rather than a return of property, and dumping them is technically conversion. In practice most neighbours will tell you to dispose of them, but ask first and the question costs you nothing.
The limit people forget
None of this overrides tree protection law.
If the tree has a Tree Preservation Order, you need written consent from the council before cutting overhanging branches, exactly as the owner would. The TPO protects the tree, not the owner's interest in it, and lopping is among the activities a TPO prohibits.
If the tree stands in a conservation area, you need to give the council six weeks' notice before the work.
Your common law right to abate a nuisance does not create an exemption from either regime. Homeowners have been prosecuted for cutting back a neighbour's protected tree in the sincere belief they were entitled to.
Practical advice
Talk to your neighbour first. You are not legally obliged to, but a conversation usually produces a better outcome than a surprise. Often they will arrange and pay for the work themselves, since a properly pruned tree is better for them too.
Cutting only what overhangs frequently leaves a tree looking lopsided and unbalanced, which is bad for the tree and bad for relations. A shared job pruned properly from both sides costs each of you less than two separate visits and produces a better result.
Use a qualified contractor for anything substantial. Cutting a large limb badly can destabilise a tree, and if it subsequently fails and causes damage, the question of who cut what becomes expensive.
Fruit
Fruit on overhanging branches belongs to the tree owner, not to you. Picking it is not one of the rights that comes with the branch being over your garden.
If the tree is dangerous
If you believe a neighbour's tree is genuinely unsafe, raise it with them in writing and keep a copy. If they do not act, the council may have powers to intervene where a tree poses a danger. An owner who has been put on notice of a defect and does nothing is in a considerably worse position if the tree later causes damage.