Wrench, wrest, wring can all basically mean to turn or twist forcibly, but they tend to vary widely in the implied purpose or result of the action.
Wrench denotes a twisting or turning with considerable force, often with an abrupt tug or yank, so that the thing affected is twisted, distorted, or forced out of position; it may stress the violence of exertion in pulling or yanking.
Wrest commonly implies a twisting or wrenching, sometimes with crude violence, sometimes with continuing deftness and dexterity, from another’s possession into one’s own. The term is common in extended use in which it implies a seizing, usurping, capturing, or extorting by such means.
Wring applies basically to a compressive twisting together, often to express or extract, but in its extended use often implies a forcing or extorting (as by urgent demands or threats) that suggests a physical wringing.