The UK is a signatory of the European Convention of Human Rights (ECHR), Article 10 of which guarantees freedom of expression. This became law in the UK with the 1998 Human Rights Act.
The text of the ECHR does explicitly mention "duties and responsibilities" and so is perhaps not as strong as the Bill of Rights in this regard, but it is wrong to say there is no guarantee to free speech in British law.
And the predicted argument about "does the UK have free speech?" has begun. I didn't say there is no guarantee to free speech in British law. I said the UK doesn't have a guarantee of the same form of the first amendment, as witnessed by the linked article where someone went to jail for tweets.
The text of the ECHR does explicitly mention "duties and responsibilities" and so is perhaps not as strong as the Bill of Rights in this regard, but it is wrong to say there is no guarantee to free speech in British law.