Let's take Glenn Miller In The Mood for an example.

I am assuming this was before the advent of magnetic tape, so how would this be recorded?

My guess they would cut an acetate master from a mixing board in the studio.

It was done to wax disc. The entire process is explained in the film "Command Performance" (1942):

https://www.youtube.com/watch?v=qdhNNaQhgv8

There were machines which would cut a blank disc, coated with celluloid or acetate, on an aluminum center. The blank discs had a second hole for a peg on the turntable, so the disc wouldn't slip under the load of the cutter. These were used in radio stations and studios in the days before tape, for one-off demo recordings or similar uses, including "air checks".

Tape became perfected by the Germans, during WWII. It was commercialized by Ampex, with help from Bing Crosby and Les Paul, and was in widespread use by the early 1950s.