As was mentioned in the comments, there is only one angular velocity $\vec{\Omega}_T=\Omega_y\hat{y}+\Omega_{z'}\hat{z'}$. This is confirmed here from some MIT lecture notes. It seems my intuition was correct.

As was mentioned in the comments, there is only one angular velocity $\vec{\Omega}_T=\Omega_y\hat{y}+\Omega_{z'}\hat{z'}$. This is confirmed here from some MIT lecture notes. It seems my intuition was correct. EDIT: If you want to use this to find the velocity of a vector, you need to cast this into the global frame first.