Two ways I know of:
Dremel (A rotory tool in case you were not sure) a slot (with a small diameter, thin dremel grind wheel) carefully across the broken stud, use flat blade screwdriver to turn out
find a small (1.5mm-2mm hex wrench) Drill down the middle of the stud with a slightly smaller drill bit then the wrench, but dont go all the way through of course. Tap in the hex head wrench and turn out.
May be other ways, but this serves me in all of my work (Motocross Race Bike rebuilds)
There is a tool called an easy out. You drill a small hole in the axle then use the easy out to unscrew it. The easy out is reverse threaded and that’s how it grabs the axle. Any hardware store will have it. You’ll need the smallest size.
IMO, a yoyo axle is probably way too small for any reasonable sized easy out, though it can probably be done with the right tools and some skill in using them. The smallest easy out I can find uses a 5/64 drill. It might work, but it’s edgy. Making the wild assumption that Auldeys use the fairly common M4 (4mm) type axle, that gives a screw diameter of approx .156 in. The 5/64 easy out is .078 in. in diameter. so this would need a hole in the screw with approx. .08 in. wall thickness. You will need a drill press or lathe to accurately drill down the center of the screw, otherwise you will probably drill off center and screw up the threaded hole.
Yeah. Good points. Didn’t say it would be easy (though it’s called an easy out). I wasn’t thinking power tools either. I have a small hand drill that you “pump” while holding the end (hard to describe) and depending on how tight the axle bit is lodged it wouldn’t take a deep hole, just enough to get a bite. It’s a nasty problem no matter how you look at it.
You really need a lathe to do the work well, considering it’s an auldey, it’d be cheaper to buy a new one.