I have a small collection of typewriters ranging from about 1910 to the mid 60s that I have repaired and restored. I needed a counterweight on the little table so I grabbed one. Then keeping with the 50s theme, I tossed in an era appropriate slide rule.
Short version: sanding, wire brushes, Scotch Brite, WD-40, Simple Green, buffing wheel w/brown compound, Mother's Mag polish, Flitz metal polish.
First all parts were washed with diluted SimpleGreen. I used an old toothbrush to get in the tight areas.
I mounted the steel shafts using "pen" jaws on my lathe chuck, using masking tape wrapped around them to protect where the jaws gripped.
At around 300 rpm I progressed from 400 to 1500 grit wet/dry paper using WD40 as a lubricant. I finished by polishing with Flitz. The screw also went on the lathe and I cleaned it with a stainless steel wire brush, followed by Flitz.
I saw no good way to remove the wooden handle, so I carefully scraped with a dull razor blade to remove paint and other foreign materials. I then rubbed it down repeatedly with Shellawax friction polish.
The metal portion of the handle I cleaned up with Scotchbrite pad followed by Mother's Mag & Aluminum polish.
Using diluted SimpleGreen as a lubricant, I sanded the curved parts of the jaws with 600 through about1000 wet/dry as seemed appropriate. The jaw faces were flattened on a piece of granite with 150 grit. I used a fine wire brushes (hand and dremel) to clean out any scratches deeper than I was willing to sand. Areas where I did not intend a high shine (the table clamps etc) I polished with green Scotchbrite.
I then buffed everything on an 8" wheel using "brown" polish, then finished with Mother's followed by Flitz. This was not quite as straightforward as written -- there was a lot of iterative repeats for problem areas as needed.