How do supernova explosions reconcile with light speed constraints? I have learned about the various types of supernovae, and both Types Ib and II involve the extremely rapid compaction or explosion of a star's core.

How does the "information" that (respectively) the Chandrasekhar limit or the degeneracy pressure have been reached propagate through the star with enough speed (yet below $c$) that the resulting collapse remains symmetric and happens in a very short time?
Interestingly enough, the limit on the time needed for such a change to propagate through the star isn't from the speed of light, but from something called the dynamical timescale, $t_{\text{dyn}}$:
$$t_{\text{dyn}}\sim\frac{R}{v}\sim\sqrt{\frac{R^3}{GM}}\simeq\frac{1}{\sqrt{G\bar{\rho}}}$$
where we've assumed that the speed of a pressure wave, $v$ is on the order of the escape velocity of the star. $R$ is the star's radius, $M$ is its mass, $G$ is the gravitational constant, and $\bar{\rho}$ is its mean density (density is not constant throughout a star).

$t_{\text{dyn}}$ can vary greatly among different stars, depending on their characteristics. It can be on the order of seconds to minutes to hours to days, and it can determine the time it takes for parts of the supernova to take place. It's a lower limit, though; large dynamical changes can take more time than $t_{\text{dyn}}$, just not less. For more information, see Section II.B of these notes.

We have to ask ourselves, what precisely is $v$? I waved the specifics away as a "pressure wave", but is there more to it than that? Unfortunately, we still don't have great models of what goes on inside a supernova - I'm sure you've heard that before - but we do have inklings of the process, which I touched on a bit here.

Essentially, as the core collapses, matter at the inner regions begins to fall in, reaching extremely high densities. Some strange things happen here, including the production of neutrinos. At first, degeneracy pressure slows the collapse of the inner star, and a shock wave forms. The so-called "supernova bounce" happens, as this wave travels through the outer layers of the star.

The shock wave is further energized by the release of neutrinos, created via electron capture (note that some have already escaped the outer layers):
$$e^-+p\to n+\nu_e$$
This adds to the energy of the shockwave, which eventually blows away the outer layers of the star. Therefore, we can see how the dynamic timescale relates to the shock wave and the time of the resulting explosion.

The exact processes causing the bounce and shock wave are not yet fully understood; this is just a basic picture.