Varying alpha news
The Webb team at UNSW recently announced new results, based on several months' analysis of new observations:
delta alpha / alpha = -(5.7 ± 1) × 10-6
This is extremely unlikely to have occurred by chance - about one in a million. However there is the possibility of systematic error, which has been addressed by the group in some published papers. I don't think the situation is anything like cold fusion: no-one is disputing the accuracy of the observations on which the claim is made, and no-one has come up with a source of systematic error which would explain the results. However, most physicists are cautious, just because it would be such a significant result if confirmed. It is also rather difficult to understand what sort of theory might explain such a variation, but if the experimental data hold up we will just have to live with it - and John Webb will probably get a Nobel Prize!
Varying "constants" could turn out to be a very fertile method to probe the fundamental theory. Any unified theory should predict relations between varying alpha and (variations in) other quantities, such as the ratio of the proton mass to the electron mass, mu = mp/me, and the gyromagnetic ratio of the proton, gp. If we have good enough measurements, it might be possible to distinguish between different candidate theories.
Varying mu?
Recently, an independent group of researchers based in St. Petersburg and Paris announced results hinting at a variation in mu:
δ μ / μ = (5.02 ± 1.82) × 10-5
from Ivanchik et al (2002). The statistical error is rather large compared to the possible variation, hence it does not constitute firm evidence.
Early Universe bounds and why they aren't important (yet)
As pointed out by CapnTrippy, Big Bang nucleosynthesis (BBN) and the CMB would be affected by changes in alpha. Nucleosynthesis is the theory that explains how the primordial abundances of light elements were created by nuclear reactions just after the Big Bang. Naturally, if alpha was different at that time, it would affect the abundances. But crucially, the data on nucleosynthesis are much less precise than the data from quasar absorption spectra.
Nucleosynthesis can rule out variations in alpha at the level of maybe a few percent, but the size of the variation claimed by the Webb group is much smaller, and would have no measurable effect on the nuclear reactions. Similarly, a combined analysis of nucleosynthesis and the CMB leads to a bound (C.J.A.P. Martins et al., 2002) of a few percent on the variation of alpha.
The variation in the early Universe could have been 100,000 times larger than the Webb results and it would still not have been seen in BBN or CMB. So there is no reason to require that alpha must first have varied one way, then returned "precisely" to its original value. One can however rule out models in which alpha varies much faster in the early Universe.
There are continuing efforts to improve the limits from the early Universe, but they will always be somewhat slippery since the observations don't measure alpha directly, but rather some combination of alpha with other parameters of the theory - which might themselves have been varying. This also applies to the bounds from the Oklo fossil reactor, which depend on poorly-understood nuclear physics as well as on alpha.
P.S. - Gilson' theory: His formula for alpha is very accurate, but his formula for the electroweak angle, once you go to the website and hunt it out, is not: why such a discrepancy? And what happens to the theory, which allows only discrete values, if alpha is varying?