UK Bill of Rights exempts government from free speech protections

No, you're right. It says "you have a right to free speech except where we've illegalized certain other speech prior to creating this document and after creating this document". One might think that a bill of rights would be a foundational document by which other laws must defer to first, but apparently not. It's just words with no power of their own. So still, so-called "extreme pornography" is illegal to produce even if all participants are consenting, and certain Japanese manga are still illegal even to possess. The Obscene Publications Act remains in full force.

There is no right to free speech in England and Wales, never mind a law aganist the government from restricting speech in all but the worst circumstances (threats, child pornography, incitement to harm, etc.) but now the government and their supporters can say "yes there is!" while pointing to a 'bill of rights' which any speech-restricting law is overriding and can override in future.