Let me replace the accelerometer by a whole smartphone because it's easy to visualize it.
Your comment that the $a_z$ acceleration was always equal to $1g$ implies that the wheel was rotating in the horizontal plane. The vertical $z$ direction contributes the gravitational acceleration $1g$ and this direction always goes along the same direction of the smartphone that you have called $z$.
To see what the other components $a_x,a_y$ of the accelerometer will show, it is convenient to work in the rotating frame associated with the wheel. In that frame, there is still the gravity $1g$ that goes in the direction that is still called $z$. But the $x,y$ axes are rotating into each other in such a way that the outward radial direction is called $x$ by the accelerometer.
There is a centrifugal acceleration – otherwise indistinguishable from gravity – which will be shown as $a_x$ while $a_y$ will be zero as it goes in the angular direction. The centrifugal acceleration is
$$a_x = r\omega^2 = 4\pi^2 r f^2$$
It is irrelevant that the speed (absolute value of the velocity) is constant. There is acceleration because the velocity is changing the direction. There is also an acceleration from the changing orientation of the accelerometer. Using the fictitious forces in the rotating frame is the simplest way to see the things.
So the accelerometer will show components
$$ \vec a = (r\omega^2, 0 , -1g) $$
If they are permuted, it means that you must adjust your idea which components of the accelerometer carry which names.
If signs are wrong, you must adapt to the conventions that the accelerometer is using about the sign (I wrote the expectation that the $a_z$ component is negative because "down" should mean a negative $z$, but the accelerometer produces whatever signs it produces: you must find the conventions experimentally). If the $y$ component is nonzero, it means that the accelerometer is rotated relatively to the simple position above. Note that if the smartphone is seemingly oriented exactly radially, it doesn't mean that the accelerometer is oriented radially. If the accelerometer resides away from the center of the smartphone, then its $x$ axis fails to be parallel to the radial direction of the wheel, and one gets the usual mixture/spilling in between the components $a_x$ and $a_y$.
As long as the accelerometer is tightly attached to the wheel, there is no Coriolis' force. The latter is proportional to the velocity of the object relatively to the rotating frame. But if the accelerometer is attached to the wheel, its velocity to our chosen wheel's frame of reference is zero.
Incidentally, it's common that some tablets have switched conventions what their accelerometers call $a_y$ and $a_z$ relatively to smartphones. So water level apps sometimes show the gravity that is rotated by 90 degrees on the display relatively to the reality etc. One must be careful about all these conventions and their possible dependence on the particular device. If the screen is allowed to switch from the portrait mode to the landscape mode, there is one more issue to worry about. Apps like that should be separately tested on smartphones, tablets, and for both, both in the landscape and portrait mode.