I have a system determined by 3 Euler angles that describe the orientation with respect to a fixed coordinate system XYZ.
The angular velocities are: $\omega_2$ (precession), $\omega_1$ (nutation) and $\omega_z$ (spin). The angular acceleration of $\omega_z$ is $\alpha_z$. The other angular accelerations are 0.
How can I find the acceleration of a point (e.g. A on the spinning reference frame xyz)? I am looking for the procedure to follow.
First of all you need to be about which reference frame you are estimating the acceleration of A. I assume you need the accelation of A wrt to the inertial frame of reference.
The most generic way is to link the acceleration of A to B. Then find the acceleration of B with respect to O.
See the following link for a starting point
The bottom line is that if you need the acceleration of P with respect to O, then you'll probably just need the following two equations:
$$\vec{v}_P =\vec{v}_O + \vec{\omega}\times \vec{r}_P$$ $$\vec{a}_P =\vec{a}_O + \dot{\vec{\omega}}\times \vec{r}_P + \vec{\omega}\times \left( \vec{\omega}\times \vec{r}_P\right)$$
