Thanks for that advice! Unfortunately my correct-fitting die appears to be a pretty basic one; no rods to screw in and all one piece. For the job at hand, I think the sliced wingnut should do the trick. I’ve never done this so I’m going to proceed veeeery slooowly. 
John Foss
Surely, that seems like a good enough approach, less aggressive than a die, I don’t know what you saw on the inter-web about it, but I’d probably put your cut at a little bit of an angle to give it a bit of a lead in so that you don’t get each ‘thread’ cutting at once. Good luck 
My pleasure. I hope you were able to repair your hub.
It’s been a struggle. M4 bolts are small! I used my (crap) hacksaw to split the wingnut, which then lead to lots of tedious time cleaning up the threads along the cut line. But it required both hands to hold the two halves, making it a very slow process to try to clean up the threads. So I upped the ante; ordered a Dremel tool (and an extra set of cutting wheels), and when that came, I was able to cut the die I’d bought online. It had three big holes around the cutting part, which allowed me to cut it to 1/3 and 2/3. This was easier to grip while rotating around the threads, but before I could finish, the 2/3 part broke at the thin spot (where I made the other two cuts). Boooo.
What next? I bought a tiny set of files. So cute! But too crude to work directly on the threads; I’d only make things worse. Meanwhile, I would try to screw on the shift button and see if it would thread on; it would just jam up. The outermost threads were all squished. No way to get around that without cleaning them up better.
First I used my tiny file to compeletly remove the threads from the first 4/5mm on the shaft. Also not easy to do; I had to make a little block to keep the file from slipping onto the threads I wanted to save. Still didn’t work; the button would jam up.
Then I realized I was messing up the threads in my button! Fortunately my die had come with a matching tap. I was able to gently use that to restore the threads in there. Whew! But I was still unable to fix those outermost, squished threads.
Chapter 25-ish: Found out my second-closest Ace store has an M4 die that looks beefier in the photo. Picked it up. Next step is to try to cut it into the 1/3 and 2/3 pieces that will hopefully be easier to hold and clean up those outermost threads. Stay tuned… [or turn of the TV in disgust as this story is going on way too long…]
I maybe misunderstood what you were going to do with the wing nut, but I don’t think you are meant to cut it into two pieces, just a single slot on one side. I think that that is then supposed to act like a ‘die’ when you thread it on to the remaining thread, cleaning out the damaged threads as a die would but less aggressively – the saw cut allows for the damaged parts to ‘cut’ out and cleared.
I’m also not quite sure why you want to cut the die you have. Do you have a die stock to hold it? Something like the one linked below (the first I found on Amazon, so nothing specific). You just put the die in that and thread it on. You’ll see the screws I mentioned in the post above in that photo, the screw by itself (opposite the group of three) should go into a split in the die, as I mentioned above, you can use that to open up the die a bit so that it doesn’t cut too much.
If you think of your die in a die stock/holder like that you can see the analogy with a wing nut.
This soap opera continues…
When I cut the wing nut it was before I bought the Dremel, so I was using my (crappy) hacksaw. It cut’s (eventually) a pretty wide swath, and it had to be opened up so I could place it on the good threads. No possibility to work it in from the end where the threads were essentially gone.
The two dies I’ve purchased (so far) are both solid; no moving parts. So again they needed to be cut to get onto the good threads. Last night I did this with my new, larger one. Unfortnately I managed to damage one of the three “cutting surfaces” with my Dremel, so I just worked the larger half of it, with its two cutting surfaces, as gently as possible as it’s very easy to lose contact with the threads.
The good news; I think I’m close to having workable threads now! It’s not perfect, but I’m close to being able to thread something on from the damaged side. I’m going very carefully and then following up with the die to keep things from getting worse. If I can make this work, this project may be over and I can hopefully get my geared 36 back on the road again.
Thanks everyone for your helpful advice! I hope to add some photos when it’s all done.
Some careful magic with a fretsaw might do the trick.
Update:
After much gentle turning of tap and (broken) die on various parts, I’ve got the unicycle put back together!
But.
Does anyone want a Schlumpf 36” that only does 1:1 or freewheel? That’s where I seem to be now. At one point while cleaning up lots of threads (riding with non-tight crank nuts for an unknown amount of time will apparently cause side effects) the “shifting spindle” pulled partially out on one side. This never happened before while I’ve worked on this uni. It seemed to pop back into place when (after panicking) I pushed it back in. But now, after reassembling everything, all I’m getting is low gear or neutral. Looks like I’m close to the finish line, but not sure where to go at this point…
And thanks to everyone above for all your kind help and suggestions!
Have you considered sending your hub to Jakob Flansberry in Canada for a Schlumpf approved/authorized repair? It’s probably not worth the cost and hassle if you’re not interested in riding a geared 36er much anymore, but if you are (or would just like to keep one in your stable of unicycles so that you have the option to ride it when desired), then I would think there is a good possibility that Jakob could restore your hub to full functionality.
I was hoping to not have to disassemble the wheel, but if no one has any insight into this neutral problem, that might have to be the next step.
Anybody else have any suggestions?
Update again:
After reading through old threads about Schlumpf hubs and sudden shifts to neutral, it recommends making sure bearing holders are not letting the knurled bearing slip, and also making sure shift buttons are not screwed in too far. I will play with those options and see where that goes…