Yes it will.
Pictured is a thin wall 12.75 collet with shaft in the chuck. Still has room to tighten up more.
No problems with the thicker collet pictured.
If you want to put a small diameter Shaft in you will differently need the thick collet
I believe it's normal for that lathe. It has delrin jaws. I also remember them using masking tape as a type of collet. Once again hoping for billiardbum to chime in.
The large chuck will close to approximately 12.5mm without collets. With the collets like shown in the pic you can get down to very small diameters because you can increase the wall thickness of the collet. Hope this clears it up.