In the United States, musicians' unions did all they could to keep recordings from displacing the work of live musicians in the radio industry; just like they're now working to keep recordings and other technologies from replacing live musicians on Broadway.
An exception were transcription recordings--recordings made specifically for broadcast--and there are many great examples of these, both by the great swing bands and by bands that became popular in the revival of early jazz styles in the 1940s.
By the 1940s, disc jockeys as individual personalities had developed, mostly playing the hits of the moment. An exception was Art Hodes, who had a program on WNYC on which he played "vintage" jazz recordings. In live radio, Orson Wells' program in 1944 featured a band of New Orleans jazz masters.