We extend our recent work on the effects of a time-varying fine-structure constant α in the cosmic microwave background by providing a thorough analysis of the degeneracies between α and the other cosmological parameters, and discussing ways to break these with both existing and/or forthcoming data. In particular, we present the state-of-the-art cosmic microwave background constraints on α through a combined analysis of the BOOMERanG, MAXIMA and DASI data sets. We also present a novel discussion of the constraints on α coming from large-scale structure observations, focusing in particular on the power spectrum from the 2dF survey. Our results are consistent with no variation in α from the epoch of recombination to the present day, and restrict any such variation to be less than about 4%. We show that the forthcoming Microwave Anisotropy Probe and Planck experiments will be able to break most of the currently existing degeneracies between α and other parameters, and measure α to better than percent accuracy
