Quade

https://www.singersewinginfo.co.uk/screw_threads This is one of the charts I was looking at.

The closest is 5/16 which according to the chart would be 5/16-24 24 threads per inch. You can verify the thread pitch by using an inch ruler over the bolt threads and count the number of threads that happen in a 1" space. 8.5mm = .33" 5/16 = .31" Singer also used 5/16x18. Your bolt looks like a finer pitch than 18. Edit - 5/16-24 is a relatively common die size meaning it's possible to make a matching bolt.

With a lathe you don't need to use a tap or die. You set the machine to cut the proper thread spacing by either changing the gears or setting the gear box to a specific speed or both. It's one of the more difficult things to do on a lathe. OP if you measure the diameter of the screw, I can try to look up the size based on a chart someone posted here. It's got singer thread sizes by diameter. A thread gauge is pretty cheap too. You could measure the pitch directly. https://www.amazon.com/ChgImposs-Imperial-Whitworth-Industrial-Measurement/dp/B07J9V9JTK

I actually hate Singer for this. I had a lathe that used oddball threads like this. It was a nightmare. I think it was like "mechanical copy protection" back in the day to try to prevent copying. I wish my machines had simple metric threads. Even standard SAE threading would be fine. I loath flat tip screws too. Was wondering of anyone sold Singer threaded screws and bolts with torx heads. It should be possible to use a thread pitch gauge to determine the thread pitch. Then someone with a lathe could reproduce that bolt.

To answer my own question. For posterity. The handwheel on the 211A and 211U comes off with the bearing. When you re-assemble, to remove any endplay, you screw in the grub screws, hold the needle bar crank on the other end, then tap on the handwheel with you palm till all slop is removed. Then tighten the grub screws. I think cleaning the shaft will make it easier to pull apart. It doesn't seem all that easy to replace the bearing. Likely you need a special holder fixture and a press to press the hand wheel off the bearing.

With a fair bit more effort than I expected, I pulled the handwheel of this machine. I removed the center screw and the two grub screws. I had to use a puller to remove it. When it finally came loose the bearing was still attached to the handwheel and pulled out of the body of the machine. Is that expected? I was expecting the hand wheel to pull off the machine pretty easily.Maybe with a couple taps of a plastic mallet. There was no bearing retainer like there is on the lower bearing. There was some dark goo on the shaft. Looked like old oil or grease. The bearing seems to be in good shape from what I can tell.

What are you trying to accomplish? That's the thing that sets the stitch length. That passage on the manual is certainly clear as mud. When you press the button on the bed, turn the crank till it locks then turn the crank forwards and back, it turns the shaft while holding disk BK. This moves an eccentric which makes that arm linkage move in a deeper or more shallow amount which determines stitch length. The other end of that shaft connects to a linkage below the main shaft which controls the needle movement so it tracks the feed dogs. This book seems like it was written by Germans. I'd suggest looking for the instructions for the 211U157A. https://s3.amazonaws.com/a.teamworksales.com/SINGER+SERVICE+MANUALS/211U157A_165A_166A_566A.pdf It seems much clearer to me.

I'd be tempted to tig weld it. My machine has visible repairs to that linkage. https://www.sewingpartsonline.com/vibrating-presser-bar-link-singer-240184 This is for the 211 which I believe shares many internals with the 111.