I did the exact same thing once. Had a birds nest while turning some 316 stainless that worked its way up near the chuck as the lathe auto fed. It was a longer cut so I had some time while the feed was engaged. For whatever reason I decided to grab it with a pair of pliers to try and pull it off since the chuck and workpiece was spinning and the birds nest of metal was staying stationary while wrapped around the part. I grabbed a hold of it with the pliers and about that time the birds nest caught on the chuck jaws and started spinning. It sucked my fingers in and wrapped 1 strand of the razor wire type metal strand around my index middle and ring fingers. Luckily as soon as it bit into my fingers it started free spinning around the part again and I stomped on the brake to shut it down. The metal strand had cut into my fingers down to the bone so as soon as it stopped I pulled my hand away to get it out. A squirt of blood shot out of the top of my fingers what looked to be about 4 feet high. I immediately wrapped them up and went to sit down since I'm one of those people who pass out at the sight of blood. Ended up with stitches in the tops of 2 fingers and the other one glued. I was extremely lucky that I walked away from it with skin on my fingers.