I'm trying to revise the Quantum Mechanical model of the Hydrogen atom, and I understand all the methods involved, including separating the wavefunction into its radial and angular parts, solving all the differential equations involved, etc. After we find the eigenvalues and the eigenfunctions and need to normalize the wavefunctions, why do we normalize the angular part of the wavefunction separately, instead of normalizing the entire wavefunction, including the radial part? Will there be any difference if we normalize everything at once? Is it a convenience issue?
It's just a matter of convenience. If you normalized the full wavefunction in a manner in which the radial and angular parts are not properly normalized and then you are looking for the spherically symmetric probability distribution along the radial direction, you'd need to integrate over the angular components, which would lead you to a radial wave function that is properly normalized, but it will differ from what you have by a normalization constant. So why not normalize that too to begin with?
Let $\Psi = R(r)\Phi(\phi)$ be the decomposition. Normalising the whole wavefunction is equivalent to integrating each component (radial and angular) over the space and dividing throughout by that number (say $N$). So now, just multiplicatively partition $N$ such that you can attach factors $a,b$ to $R,\Phi$ such that $ab = N$. Now I leave it to you to show that if you go the other way (i.e. find $a,b$ first for the components) then $ab =N$ that is the overall normalisation is unique. This way, we can directly normalise the components and gurantee that $\Psi$ is normalised.