Assume in the given diagram, $\theta = 0$, for sake of simplicity
Then, I understand there exists an initial angular momentum in the y-direction and a torque in the X direction of the gyroscope. This causes the angular momentum to chase the always perpendicular torque and hence it precesses a circle. However, this rotation in the yx plane gives it an angular momentum in the z-axis that did not exist before, and to my understanding, no torque acted on it to make it so. Where does this angular momentum come from? What applies the torque needed for this?