I'd suggest the descriptor
Old time radio voice
It accurately captures a certain confluence of time, technology, and culture.
Edit: Except I can't because this is set in the past. So how about "radio voice"? There wasn't an established video style yet, so they were borrowing the radio style. If I were watching, I might think, "oh, they sound like radio presenters." Or maybe I wouldn't, because what else would they sound like? ¯\_(ツ)_/¯
Alternatively, you might reference a famous newsman of the time. It might or might not be before Walter Cronkite or Ed Sullivan, but Walter Winchell or Edward R. Murrow seem appropriate.
Edward R. Murrow knock-off
When it's said that someone's a Cronkite wannabe, even if you don't know exactly what Cronkite sounded like, you know it's a reference to the defining newsman of his era, so you automatically conjure visions of 60's newscasts.
If you don't know who he is, well, have you ever noticed that when you hear a reference to a person of a bygone era, it has a certain legitimacy above a phrase from a bygone era? It's more anchored in its time and place. You Google him and find out who he is in 10 seconds, and think to yourself "that author's pretty smart with his 40s newsman references".