Karl,
As was mentioned earlier, the name should be stamped into the end of the bed and the headstock frame, look there to see if you can read it.
It appears to be an American Watchmakers Lathe "WW style" with the headstock and tailstock indexing on the outer "V" part of the bed, but other lathe manufacturers used this "WW style" also.
Typically these have an 8 mm collet chuck opening in the spindle, you can measure that to confirm.
As for restoring, the lathe was nickel plated when new, not chrome plated, and this is rather difficult to have done today, as most places stopped this process because of restrictive EPA regulations.
To clean it up you can take it all apart and soak it in a de-rusting solution like Evapo-Rust, or a dilute solution (2 tablespoons per gal.) of Rust Out and water Watch it every hour as it may start to turn dark black if left in too long. Then use a soft bristle SS wire brush to clean off the softened rust or #0000 steel wool.
If you want to polish it you can use 1200 - 6000 grit wet/dry paper to smooth out the castings and plating edges so they look smooth and them polish with Simichrome Polish or similar metal polish.
As for missing parts, ebay would be a source, otherwise, you can make them if you have access to another lathe.
That is the greatest feature of a lathe, it is a tool that can make itself.
These lathes are smooth operating and can be restored to turn very accurate parts if you have the patience and skill to work a hand graver or can find a cross slide to fit the lathe bed at a reasonable price.
If you have more questions, email me offline at:
ziegedw@gmail.com
Dean Z.