Company, firm, and business have specific meanings.
Given BBC is UK based, a company would mean a separate legal entity to its members (a 'corporation' in US).
A firm means a group of people in business together, either a company of more than one member or a general partnership.
And a business is the true catch-all.
I doubt these businesses going back hundreds of years are incorporated entities. Back in the day those took an act of the state to form. The first KK in the country (Inc. equivalent for those in US, or Ltd. for UK) was only formed in 1873.
And I doubt businesses have been general partnerships continuously for hundreds of years -- since these desolve automatically when the membership changes e.g. by a death.
Especially to journalists, these details should matter.