If I understand you correctly:
You describe a gyroscope wheel with the spin axis perpendicular to the Earth's axis, mounted in such a way that there is no freedom to pitch, but the swivel axis is connected to electric generators.
To establish names for directions of rotation I add an image of a gimbal mounted gyroscope
Spinning of the blue gyroscope wheel: spin axis
Motion of the red frame relative to the yellow frame: pitching
Motion of the yellow frame: swiveling
As I understand your description:
A setup where there is no possibility of pitching motion. Compared to the setup in the image: regard the red frame and the yellow frame as a single solid unit, a perfectly rigid structure.
For comparison let me first describe what would happen if the setup does allow pitching motion but with brakes that you can quickly release.
You spin up the gyroscope wheel, initially using the brakes to prevent pitching motion. As the gyroscope wheel spins up the combination of the spinning and the co-rotating with the Earth along the swivel axis gives a tendency to pitch. Since the Earth's rotation is so slow that pitching tendency is very small, but with a sufficiently high spin rate you can eventually achieve a significant tendency to pitch.
When you release the brake of the pitching motion you can attempt to harvest energy from the pitching motion. Any harvesting that you achieve stops when the spin axis has become aligned with the Earth's axis.
Now the case with no freedom for pitching motion:
As you spin up nothing will happen.
Prior to spinning up the swiveling motion of the gyroscope was co-rotating with the Earth's roatation, and all the way while spinning up the swiveling motion will remain co-rotating with the Earth's rotation. Because of the tendency to pitch stress in the supporting structure will mount, proportional to the spin rate. But this setup is described as perfectly rigid, so it can entirely prevent pitching motion.
[LATER EDIT, 3 hours after initial posting]
I think I need to clarify my answer.
The key factor in gyroscopic effects is motion.
When you have a spinning motion, and you add another rotational motion, along a different axis, then in response the wheel will move along a third axis.
The simplest case is when the secondary rotational motion is perpendicular to the spinning motion. Then the response is along an axis perpendicular to both of the first two axes.
But then that third rotational motion has its own effect, and it's easy to misinterpret that.
So take the case depicted in the image. You spin up the wheel, and then you put your fingers on the outer frame to try and swivel the gyroscope wheel. In response the wheel pitches, and that pitching motion results in a tendency that opposes your attempt to make the wheel swivel.
So it feels as if your attempt to make the wheel swivel is opposed by the spinning wheel just like that. If the spin rate is very high a very slow pitching rate is sufficient to give a palpable opposition to the attempt to sustain swivel. It's easy to think of the pitching motion as insignificant, but it very much isn't.
It is difficult to suppress pitching motion altogether, the usual mountings aren't set up for that. Those are factors that contribute to misinterpretation of how a spinning gyroscope wheel responds to being nudged.