From my understanding, I'd like to point at a few things. (bear with me)
If superstring theory is correct and all, magnetism could never stop existing. It's just how string theory says the universe should work.
But if we let magnetism stop, you might want to note that we have 4 forces in the universe:
1. Gravitational Force
2. Electromagnetic Force
3. Strong Force
4. Weak Force
You might not have heard of the last 2, so I will briefly explain.
Inside an atom, all of the protons are grouped at the center. Also, they have positive charges. Putting that in mind, electromagnetic forces should push them apart, like when you try to stick two magnets together from the positive side, they repel. But strong force counters this and holds the nucleus together (and is stronger).
Weak force is a weak atomic force that causes radiation. It is usually overpowered by other forces.
So when we remove electromagnetic forces (magnetic = electric), a few things should happen. First of all, electrons (-) and protons (+) should say hasta la viesta because + and - attract due to electromagnetic force. But now that's gone, so oh well.
You'd have some major radiation due to all of these electrons flying everywhere, which should kill you (that's somewhat what radiation is).
You'd have, I think, protons still sticking together and neutrons in there as well. The catch though is that they will be more dense with little to stop them from getting denser.
You might not even have neutrons at all: neutrons form from electrons and protons forced together, which has more mass and neutral charge. But what holds together neutrons? I think that'd be electromagnetism, which is gone.
However, neutrons may still exist due to gravitational pull inside of massively dense objects like neutron stars.
I do believe that at this point, if most of what I have said is true, then matter wouldn't exist with atoms. However, if matter does stay in some deranged way, then you might come to realization that it doesn't affect the main pillars of physics too badly. Relativity relies, indeed, on the speed of light, yes, but other than that, it should still work. Quantum Theory should also work, it would just affect some of our calculations and such. And chaos theory should still be itself, as chaos theory, and generally it should prove to work 'normally', whatever that's supposed to mean. If I'm wrong about this, do excuse my understanding because it is not extremely thorough.
Interestingly enough, I'd like to point out that light relies on electromagnetism to work the way it does and that without it, light may not even exist. In fact, electromagnetic radiation may not exist. Unless the universe comes up with a substitute, as I will discuss.
Without light, what do we have? "I think, therefore I am" -Descartes
Without light, you can't observe the universe in a fundamental way. Yes, we have the blind, but this is different. In a way, quantum theory's uncertainty principle relies on this idea of what we know and what we don't. Being unable to observe the universe in one of the main ways that life does, the uncertainty principle puts some majorly weird quantum mechanics into the world. For example, you might place a cup on a table, but without being able to observe the cup, it may be anywhere. (even off of the Earth or further) (Related to the problem of something going through two doors when you don't look)
But besides that, we have string theory. Essentially, the universe is an orchestra, with a set of notes to play. The universe may play any notes in any order to produce anything, but there is a limit. It only gets some notes to use. That is, it creates the 4 forces named at the beginning and only those forces. It creates matter as we know it, but it is limited to only certain elements, certain types of matter. The grand orchestra of the universe has its rules, according to string theory.
But when the universe decides to stop with electromagnetism, it means it full stop on certain notes, or chords, or whatever. This means the effect will be catastrophic. From what I understand, your end spells no beginning as far as new notes for the universe's orchestra.
What this means is that many of the universe's fundamental properties simply disappear altogether, related to electromagnetism in a direct way or not.
So the effect is a lot stranger than first appears.
As a side note, the 4 forces contain symmetry, something that would no longer exist without magnetism, which may have its own effects on the universe.