I just did a '57 radio for a guy. It was the regular set, and not the Brougham model. The car is a converible, but the radio was the same for closed cars or convertibles in 1957.
There are two speaker lines coming from the receiver chassis. In both cases, there are two conductors to the speaker; one with a connector to the front speaker, and two others going to the rear speaker. I recall that the rear speaker wires are brown and black. On the set I worked on, some blacksmith had cut the rear speaker wires, so I don't know what the termination is for them. Somehow, they would connect into the body harness.
If you bench test the set before installing it in the car, best to clip on a test speaker to either pair of the speaker wires. Put the front-rear speaker control mid position, and either pair of wires should give you sound.
If there is no action from the speaker leads, it will be necessary to go to the power amplifier chassis (the one with the vibrator), and see if there is signal right off of the output transformer. I can supply you with a reprint of the Delco service folder, if it'll help. My address is in the membership directory.