Timing affects how well a tree recovers from pruning, how much it bleeds, and how vulnerable it is to disease. For most trees the answer is winter, but the exceptions are the ones that cause damage.
The general rule: dormant season
Late winter, from roughly November through to early March, is the standard window for most deciduous trees.
The tree is dormant, so it is not losing energy through active foliage. With the leaves off, the structure is visible and the arborist can see exactly what they are cutting and how the crown is balanced. And wounds made late in dormancy begin sealing quickly once growth resumes in spring.
The exceptions that matter
Plums, cherries and other Prunus species should be pruned in summer, not winter. Winter pruning exposes them to silver leaf disease, a fungal infection whose spores are abundant in the cooler, wetter months. Pruning in dry summer weather, when the tree is growing strongly and can seal wounds fast, greatly reduces the risk.
Apples and pears are pruned in winter for structure, and can be summer-pruned to restrict growth on trained forms.
Birch, maple and walnut bleed heavily if cut in late winter or early spring, as sap rises. It rarely kills a healthy tree but it looks alarming. Prune these in late summer or autumn instead.
Evergreens are generally best in mid to late spring, after the worst frosts.
The rule that overrides all of this
Nesting birds. Between roughly March and August, trees and hedges may hold active nests, and it is an offence under the Wildlife and Countryside Act 1981 to intentionally damage or destroy the nest of a wild bird while it is in use or being built.
This does not ban summer tree work, but it does require the tree to be checked first, and work to stop if an active nest is found. A reputable contractor will check as a matter of course.
Bats are also protected and roost in mature trees with cavities and loose bark. Where a tree has obvious cavities, a bat survey may be needed before major work.
When timing does not matter
Dead, diseased, damaged or dangerous wood can and should be removed whenever it is found. There is no benefit in leaving a hanging limb over a garden until February.
The same applies to anything creating an immediate hazard.
Permission still applies
Timing is a horticultural question, not a legal one. If the tree has a TPO you need written council consent regardless of season, and in a conservation area you need to give six weeks' notice. Neither requirement relaxes because it happens to be January.
Factor that lead time in. If you want winter pruning on a protected tree, start the application in autumn.
How much to take
Less than you think. A common instruction is to reduce a crown hard to let in light, but removing too much foliage in one operation stresses the tree and often triggers dense regrowth of weak, upright shoots.
A reduction of more than about a quarter of the crown in a single season is a lot for most mature trees. A good arborist will tell you if what you are asking for is more than the tree should take, and will suggest staging it over two seasons.