I think 'radio voice' here is more than learning some idea of a a regional/neutral? accent. A cohort had been an announcer.. periodically we'd say, "OK Don - give us that in your radio voice.." His voice would drop a few semitones and slow. People who teach diction, like 'enry 'iggins, employ a large vocabulary which includes 'projection' and other intricacies of voice production (singers also know all about that.)
OTOH, my perception of the BBC newsreaders heard across the years - is not so much about such a special projection as - some consensus that the (present-) Queen's English style [high English == associated with the Peerage more than the Commons] is officially the accent to be emulated. That's not a special way of projecting the voice, though - as I hear and compare.
I used to play with that a bit too, when making certain PA announcements (especially dire ones like, "This is Not a drill - we ARE clearing the building! ... NOW! KID!") As with all masquerades of pomp & circumstance - fun.
I seem to have misplaced my General Patton approximation, though :-/