No we don't. We have what is referred to as an "uncodified constitution".
https://en.m.wikipedia.org/wiki/Uncodified_constitution
It is a collection of laws and conventions, but there is nothing set up as an overarching set of rules to guide the country. If something were to happen that was deeply unpopular with what the majority of the country feels "makes us British", there is little we could do about it.
Successful court cases against the government have usually been because the government of the day forgot to pass the law that gave them the power to do whatever move they wanted to make. A constitution change is a much bigger deal.