Constitutional monarchy doesn't really mean that, Sweden and the Netherlands are constitutional monarchies, and they have a proper constitution.
The UK doesn't really have a constitution but a collection of old laws, statues and treaties, such as the English Bill Of Rights, the Magna Carta, and the Act of Settlement; and the concept of Parliamentary Supremacy.
Under Parliamentary Supremacy anything can be changed by a mere act of Parliament, it is only because of convention and tradition (and of course threat of electoral defeat) that stops Parliament from imposing things like the death penalty for speeding.