If two gyroscopes were to be started with parallel angular momentums, linked by a rotating tube (one perpendicular to the axle), then a second pair added that is the exact reverse such that it has opposite gyroscopic precession, the two linked such that they may not move relative to each other, and then each gyroscope moved by the rotating tube in the opposite direction of the one on the opposite end for each pair, where would the energy go that was used to force the gyroscopes to move away from their original orientation? I can add a drawing of the apparatus if it would help.
In general, energy is not required to change the direction a gyroscope is rotating. Similarly, it doesn't require any energy to redirect an object with linear momentum either (we don't consider there to be energy transfer when a billiard ball caroms off a cushion).
In the case of the bouncing ball, there's no energy transfer because we consider the wall/cushion to be fixed in place. So the force applied happens over zero distance.
In the case of the gyroscope, the gyro precesses at right angles to the applied force. Again, the cross product of the force and the distance moved is zero.
You raised a question in comments about a spring pushing on a gyroscope. I don't think I understood the geometry. A diagram might help.
Your diagram shows a spring pulling down, so there is force beyond just the weight of the gyroscope. We could do the same thing by adding a weight to the unsupported end of the gyroscope.
So the statement that it doesn't take energy to move the axis is correct. But it does take a bit of energy to start moving it. (just like it takes some energy to accelerate a ball, but none to keep it rolling, sans friction). This energy comes from a slight drop in the gyroscope.
The gyroscope falls slightly. As it falls, it picks up speed sideways. The greater the downward force, the faster it moves sideways. The faster it moves sideways, the more it counters the downward force until it reaches an equilibrium. Lowering the force would reduce the sideways speed.