The BBC aren't really independent of the government. They like to claim this and mostly get away with it, because British governments tend to be lenient with them and don't interfere. But they depend on taxes for the bulk of their income, their existence is defined by law and the government appoints the person who runs it. A change of government could completely change the BBC tomorrow and there'd be nothing anyone working there could do about it.
The existence of any media organisation (or any corporation) is defined by law; a change of government could in theory completely change any organisation in the UK.
But yes, "the Chairman and the non-executive members for the nations are appointed by HM The King on the recommendation of Ministers."
In reality, I suspect the ownership structure of a media organisation matters less than the ideologies of its directors.
