ECHR Article 10: "Everyone has the right to freedom of expression. This right shall include freedom to hold opinions and to receive and impart information and ideas without interference by public authority and regardless of frontiers. This Article shall not prevent States from requiring the licensing of broadcasting, television or cinema enterprises"

(yes, ECHR is still UK law at the quasi-constitutional level, despite the huge pressure against its application by non-nationals)

On the other hand: this isn't actually a free speech issue, it's an "access to classified information" issue. The MOD is simply refusing to say what it's doing. I'm sure the US DOD doesn't give out information on classified operations to random congressmen either.

Probably wouldn't make much difference. British freedom of speech is not actually that bad in general, RSF ranks the UK significantly higher than the US in this regard: https://rsf.org/en/index

This article goes into it in more detail, but in honesty, the first amendment does not actually always stop Congress from passing laws restricting speech under various circumstances.

https://scholarworks.law.ubalt.edu/cgi/viewcontent.cgi?artic...

We have 'freedom of expression' and case law, but these things end up having to go to court, so I think it being written in a constitution would be better.

That would require the repeal of a fair tranche of the terrorism legislation. It is illegal in the UK to "express an opinion or belief that is supportive of a proscribed organisation" in the Terrorism Act 2000, for example.

Britain has "freedom of expression" with some very specific limitations (lese majesté for instance).