The mu and alpha constant have changed a bit in the history of the universe (about 13.7 billion years). How much would other constants have to change in order for everything to start breaking apart? ie, planets cannot revolve around suns and galaxies start to malform due to gravitational constants changing, atoms breaking apart because strong forces are changing, fusion stops because of weak forces changing, etc.
First of all, no reputable scientist is claiming that “The mu and alpha constant have changed a bit in the history of the universe”. There are some results which suggest that, but the error bars are still large enough to be consistent with them being true “constants”. They may well have changed, but we don’t really have solid enough evidence yet to say so with any sort of certainty.
That said, I would imagine that your question would depend a lot on how fast they changed. You could construct atoms with a wide range of mu and alpha. On the one end of the scale, positronium (an “atom” consisting of an electron and a positron, instead of a proton) has been observed to exist, and that’s equivalent to a hydrogen atom with mu = 1. And on the other hand, tritium effectively has a mu three times the normal value, and that’s chemically equivalent to hydrogen, too. Quantitatively, these three atoms have different fundamental frequencies in their spectra, but qualitatively, they all behave the same.
The case of alpha is a bit different, since things go a bit wonky if alpha >= 1 (at least, we don’t know how to formulate electromagnetism, for such a case). And if it had a significantly different value, some phenomena would be more or less significant than they are now (for instance, for alpha much larger but still smaller than 1, photon-photon interactions would be much stronger). I think that you could safely decrease alpha by a few orders of magnitude without too many qualitative changes on the macro scale, but if you increased it more than a small amount, a lot of things which are not significant in our world would in fact become significant.