Assuming the nature of doubt is in approach.
COME can be used and followed by differentiation w.r.t. t which is one of the standard approaches in SHM.
Potential energy change takes place due to the movement of m and m'. KE change takes place due to the rotational kinetic energy of the pulley+m system and \(\frac {1}{2}m'v^2\) energy of m'.
After differentiation one can find \(\frac {d\omega}{dt}\) in terms of \(\theta\) after making the approximation of small \(\theta\). Term containing \(\theta\) is what should be of interest and \(sin(\theta+\alpha)\) expansion may be required.
The answer as obtained above involves m'. However, you have one more condition in the beginning which can be used to remove m' if you like.
In the beginning when the system is at rotational equilibrium, \(m'gR=mgRsin\alpha\).