Not sure if you can classify the BBC as a "private business". It's a company which has a Royal Charter, and its governing body (the BBC Trust) is composed of twelve members appointed by the Queen.
When it is almost impossible to tell the difference between state, charity, business and government you have Mussolini's vision perfectly enacted. Society now serves to enhance the power and prestige of the state rather than the other way around.
But I disagree that it's almost impossible to tell the difference. The BBC is an extremely different company in many fundamental ways.
I think that if we were to consider the BBC as a signal of Fascism, then the term loses all meaning because even the states criticized by Mussolini had state controlled companies, from the what he called supercapitalists to the socialists.