@Laundress, yes, the service manual for Kenmore wringers on here covers this 1957 model. Also, yes, it's a Whirlpool OEM.
The pump seems to turn fine. It's just a matter of getting it to move properly so it will hold tight to engage with the motor.
Thanks for the insight on the timer. If I can get it going then that'll be cool. Currently the agitator goes even when it's in the off position.
As for the polish and whatnot, I've seen someone use some Mother's chrome polish on the wringer and whatnot. For the rest of it, yes, some good cleaner wax I'm sure would do a fine job. I intend to give it a bit of touchup or a new paint job entirely with some Rust-Oleum appliance epoxy on the skit of the machine and lid. The porcelain doesn't seem to need any touchup. I'll get some better pictures the next time I'm working on it.