The current trough cascode transitor is fixed by the current mirror. So, there is no effect of the noise, the current stays the same (assuming output resistance of the current mirror transistor large enough).
Try to change the bias voltage at the gate of the cascode transistors, is trhe same thing. Constant current => Vgs constant => the noise at gate appears at the source of cascode transistors with the same amplitude => the source is connected to a drain => high resistance => no variation in current.