The speed of light does change physics, as does any fundamental constant. Expressing a speed in different units does indeed yield different numbers, but that is not equivalent to a change of speed.
Changing alpha means that electromagnetic force get stronger or weaker, thus the atoms still exist but with different orbital levels. Considering how intricate the orbital levels of atoms are, it's not shocking that a tiny change in alpha completely redefines chemistry, but chemistry would still exist.
The point is that all of the speeds we measure are actually measured in units of the speed of light, if you go right down to it (seconds and meters are defined in terms of the speed of light). So, the value of the speed of light doesn't really matter - which is why c=1 is often used in many kinds of physics.
The speed of light does change physics, as does any fundamental constant. Expressing a speed in different units does indeed yield different numbers, but that is not equivalent to a change of speed.
Changing alpha means that electromagnetic force get stronger or weaker, thus the atoms still exist but with different orbital levels. Considering how intricate the orbital levels of atoms are, it's not shocking that a tiny change in alpha completely redefines chemistry, but chemistry would still exist.