The plum tree needs a bit of a cleanup, particularly the long leggy shoots that have gone straight up from previously pruned areas, so I had a look at when the best time to do this is. Quoting various gardening web sites, "pruning is done in winter when trees are dormant", "the best time to prune fruit trees is late winter into early spring", "prune in late summer after fruiting has finished, however it can be done in early-mid autumn if necessary", "best pruned in spring or summer", "best pruned in midsummer", "prune immediately after harvest", "plum trees should only be pruned in their growing season", and many more. So the best time to prune is definitely winter, or spring, or summer, or autumn, depending on who you ask.
And in terms of the need to do it: "leaving any kind of plum unpruned will not do much, if any, harm", "it is important to prune each year". So pruning is both important and not important.
Homeopaths have got their act together better than this.