If you are careful and have enough room or long enough chisel or old screwdriver you can make notches to get the socket or lug not removal tool something to grab on to. Also depending on the wheel if you can get some heat on the stud itself directly, then try to loosen it....Just some thoughts, good luck!
Make a notch in the face of it with a chisel and use air hammer to spin it. I do it fairly often on rounded nuts or when owners loose the key socket for locking nuts like the round McgGard ones.
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John Paige
70 Firebird Esprit, 400 TA clone type "The 14 car"
lab-14.com
I lost the key for the locking lug nuts on my 72 C10. Found a pipe with similar diameter as the nuts and tack welded it on. Spun it loose with a pipe wrench.
Yikes! It looks like you've already marked the wheel up. If it were me, I'd drill the studs, at this point. It will take some work, but you can drill the studs out with a good, high speed corded drill, and a quality bit. It looks like the end of the studs are concaved, so centering the studs should be fairly easy.
Ouch, I would try to see if you could hammer on a 12 point socket on there. Trying to drill the stud will be a pain and trying to cut the nut will only put more hurting on that wheel. Are you using an impact gun? High speed spinning is the key.
Also put undamaged nuts back on the other studs and torque them to down. It will take pressure off of the messed up ones.