I made the same lathe and chuck a couple of years ago. The difference is I made the chuck body from one piece of steel and the jaws were made from gauge plate and hardened and tempered. I also fitted miniature ball bearings to spindle and a collet set plus an electric motor with variable drive. Sorry about leaving the chuck key in but it looked better for photo.
I have since made a cover for the motor to keep out swarf and it looks more in keeping.
David