  The success of primordial nucleosynthesis as a cornerstone of the hot Big
Bang model has been limited by the long-standing lithium problem. Recent work
presented a self-consistent perturbative analysis of the effects of variations
in nature's fundamental constants on primordial nucleosynthesis for a broad
class of grand unified theory models, showing that such models provide a
possible solution to the lithium problem, provided the value of the
fine-structure constant $\alpha$ at the nucleosynthesis epoch is larger than
the current laboratory one by a few parts per million of relative variation.
Here we extend the earlier analysis, focusing on how this preferred value of
$\alpha$ is affected if relevant cosmological parameters are also allowed to
vary--specifically focusing on the baryon-to-photon ratio, the number of
neutrinos, and the neutron lifetime. We rephrase the lithium problem in terms
of the values of these parameters that would be needed to solve it within this
class of grand unified theories, thus obtaining values that would disagree with
the results of other experiments by several standard deviations. Using these
experimental results as priors in the analysis, we find that a larger value of
$\alpha$ is still preferred, confirming our previous results. By excluding
lithium from the analysis, we also obtain upper limits on possible variations
of $\alpha$ at the primordial nucleosynthesis epoch. At the two-sigma level,
these are $|\Delta\alpha/\alpha|<50$ ppm without nuclear physics, cosmology, or
atomic clocks priors, or alternatively $|\Delta\alpha/\alpha|<5$ ppm if these
priors are used. While the simplest solution to the lithium problem is likely
to be found within observational astrophysics, our work shows that varying
fundamental constants remain a viable alternative.
