Element generation in the Big Bang is a fine laboratory to test possible variations of the fundamental constants of the Standard Model. In this talk, I concentrate on the electromagnetic fine-structure constant αEM. First, I study the dependence of primordial nuclear abundances on fundamental nuclear observables such as binding energies, scattering lengths, neutron life- time, etc. by varying these quantities. The numerical computations were performed with four publicly available codes, thus facilitating an investigation of the model-dependent (systematic) uncertainties on these dependencies. Second, the dependence of the primordial nuclear abundances as a function of αEM is studied, keeping all other fundamental constants fixed. For that, the leading nuclear reaction rates, in particular the electromagnetic contribution to the neutron-proton mass difference pertinent to β-decays, are updated and certain approximations made in the literature are avoided. In particular, the temperature-dependence of the leading nuclear reactions rates is included. The systematic uncertainties are assessed by using five different publicly available codes for Big Bang nucleosynthesis. Finally, an outlook is given on the calculation of the quark mass dependence of primordial nucleosynthesis.