if the fine-structure constant had different values at different places of space (which I don't believe for many reasons), then the size of the atoms e.g. in Planck units would depend on the location, too. Correspondingly, electrons and protons would "sit" into Hydrogen atoms at slightly different moments and different places would perform recombination - the birth of atoms when the Universe became nearly transparent and when the CMB was born - at slightly different moments. At different moments, the temperature was a bit different.
So a part of the observed anisotropy of the temperature of the CMB radiation could be blamed on the non-uniformity of the fine-structure constant in the times when the CMB was created. However, physicists would always think that the non-uniformity of alpha didn't arise "just for fun" but it was probably driven by something else. So they would ultimately reparameterize the influence of a variable $\alpha$ to the influence of something else, anyway.
You can't see any alpha-effects in the detailed structure of the microwave background itself - except for the overall temperature discussed above. It's because the CMB radiation is a black body radiation which doesn't depend on the character of objects at the time when CMB emerged. Photons are massless and their properties - except for interactions with charged objects - don't depend on any alpha, either. In fact, the CMB radiation is the most accurate natural black body radiation we have ever observed.