How likely is getting gangrene in your hands?

Small: Unlike the foot, there are two arches delivering blood flow to the fingers and hand. There is excellent collateral circulation in the hand as well. To develop gangrenous changes, all the blood flow would need to be shut down so low as to allow ischemia and gangrene to start. That is not likely. (it can happen though, like in frostbite...).

Very low: Barring severe vascular problems or a bizarre infection.