Many (most?) western societies have a similar concept for civil and criminal law with common law jurisdictions, where precedent is used rather than an explicit, exhaustive legal code. Effectively, the UK's constitution is to written constitutions as common law is to civil law.
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.
We don't have the state mechanism. You could argue the four nations could serve a similar purpose, though there's a debate about how democratic that is when England makes up something like 85% of the UK population (and doesn't have its own legislature).