I missed that programme, but always wondered why the apostrophe was in one
but not the other. Why is Barons Court so named? (Oh, and just to
confuse
matters, the name on the street signs nearby is "Baron's Court Road"!)

Well ignoring whether or not it ought to have an apostrophe, I just had a
look around and it seems the suggestion was that it was an invented name for
something like a housing development. Similarly you get Kingsbury which is
an old name, say Saxon or something like that, and Queensbury which is from
modern times.