I don't know why I started humming this today.
I always thought of the phrase 'backroom boys' as an RAF thing. The Oxford Dictionary of Phrase and Fable attributes it to Lord Beaverbrook in 1941.
The film Destry Rides again in which Marlene Dietrich sings The Boys in the Back Room is from 1939 so the phrase is older than I (and the dictionary compiler) thought.
Except, the Backroom Boys that Scots band Nazareth sung about were gamblers so perhaps that was what was what they were up to in back of the Bottleneck Saloon. Nothing to do with radar at all.