There's many of times I have to stop the lathe in a cut, shut the power off, get a pair of H-F needle nose pliers and pull those nasty stringy shavings that wrap around your work while turning, off the workpiece, then continue cutting.
On the lathe, I use chip brushes to get the chips to the chip pan, then use a small flat blade shovel to move them to the garbage can and or heavy leather gloves to handle the fine metal chips. And of course, the gloves go back to it's storage place once done and never use when the lathe is turning.
For removal of chips from the mill, a combination of chip brushes and the shop vacuum cleaner works best! And nothing wrong using it on the lathe, too!