[interestednovice]

You are factually right, but I think this is one of those cases where oversimplified but understandable terminology is justifiable over more accurate yet opaque jargon. Speaking to a general audience, I would say that unwritten is an acceptable substitution for uncodified as the audience will understand the term and I doubt many viewers will think that all of British constitutional law is literally not written down anywhere. I've also seen the argument before that unwritten in these terms refers to a single constitutional document having not been written, rather than nothing have been written down.

It's like I wouldn't expect a presenter to pull anybody up on using the term 'First Past the Post', even if such a term is looked down upon by political scientists for being an outright misnomer (Single-Member Plurality voting is a relative system, there is no post that candidates have to cross to win).

Technically, it is also unwritten.

There is no document called “The UK Constitution’”. Our constitutional settlement is, in fact, unwritten as the law that governs the country is a mixture of traditional procedure under the prerogative powers (excised by, or on behalf of, the monarch via the Royal Prerogative), laws enacted by Parliament (which technically derives it’s authority from the monarch “appointing a government”; traditionally done after an election but legally permissible at any time) and case law precedent. The prerogative powers, as uncodified in any document, are not written down anywhere - merely exercised. Exactly how they can be used is a point of debate even amongst constitutional scholars as they are only bound by precedent, not written law.

All of these things put together make up our constitution, so it is correct to say that it is unwritten - both because it is not a single written document, like many other countries; and also because many aspects of it are not formally written down at all.

If you think that, even then, you would prefer the word were not used and it be described as not codified, the point above (that simplification to allow the majority of layman to understand what is essentially meant, even if the wording is not legally identical in meaning) is a valid one.