In the absence of a written constitution whatever is unconstitutional is the ad-hoc interpretation of the Justices (or whatever they are called in the UK). There are arguments in the US about strict constructionists vs. judicial activism in regards to justices and judgements.

How do those debates and decisions happen in the UK? Do they just go with whatever the Judiciary deem to be constitutional zeitgeist of the land?

Don’t they use past decisions as juris-prudence? With centuries of constitutional decisions you start to have a good amount of records to base your judgement on.

But generally speaking I feel that the “interpretation of the constitution” where judges have the responsibility to interpret century old documents is a very US thing, other democracies generally have a more recent constitution, and see it as a living document.

Largely they don't. It's simply not a significant part of UK political discourse, unlike America where many important rights (even interracial marriage!) are the result of court decisions.

It has only picked up in a couple of areas: the conflict of ECHR especially right not to be tortured and right to family life with UK immigration law. And of course around Brexit.

They are the final court of appeal, but they can't strike down primary legislation because Parliament is sovereign. They can strike down secondary legislation and legislation that may be contrary to the Human Rights Act (with the agreement of parliament in many cases). They can make life difficult for the Government, but Parliament can always make new laws. I think the UK system is more democratic because it does not give the judiciary primacy over elected representatives.