In the UK it's enshrined in law as "freedom of expression", which is a bit wider than freedom of speech, though is inclusive of it. Freedom of speech is a term that's often bandied about without understanding what it is, or how it applies.
Our right to freedom of expression goes back to the Universal Delcaration of Human Rights drafted by the UN in 1948, which is a pledge by the state, to it's people to protect it's them against it's own tyranny and oppression. Most recently it's enacted in the 1998 Human Rights Act as Article 10, and this law prevents government interference with our freedom of expression.
However, it is not an absolute right, and may be limited. So freedom of expression does exist, but it is curtailed by other laws. We can see the precedent for this in defamation, libel, use of threats, incitement to violence etc. So not all expression is protected, particularly if it violates the rights of another.
I think that's quite important to understand, as a private individual can not violate your freedom of expression, nor can a private company. The former may fall foul of other laws in doing so, and the latter is largely governed by contract law. Only the state can limit your freedom of expression, and whither or not we've got the correct balance is always, and should always be up for debate.
Lineker's case is intersesting, as I think an argument could be made that the BBC is the state, and therefore may well be falling foul of Article 10 of the HRA by seeking to limit the expression of it's employees, when it's on their own time. Though I don't know enough about corporations to say if that's an argument that holds any weight. But if state run schools and institutions can fall foul of the HRA, I don't really see why that would be different for a state broadcaster.
And the UK does have a constitution, though it's spread across many documents, and is not all written down. I don't really want to dive into constitutional law, as it's very complicated and pretty dry, but the Magna Carta, the Bill of Rights, the Act of Union, the HRA, etc along with various parts of common law, and convention form our constitution.
Anyway, that's just some of my random musings on the matter.