  Higher-dimensional theories have the remarkable feature of predicting a time
(and hence redshift) dependence of the `fundamental' four dimensional constants
on cosmological timescales. In this paper we update the bounds on a possible
variation of the fine structure constant alpha at the time of BBN (z =10^10)
and CMB (z=10^3). Using the recently-released high-resolution CMB anisotropy
data and the latest estimates of primordial abundances of 4He and D, we do not
find evidence for a varying alpha at more than one-sigma level at either epoch.
