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There's a famous saying in science fiction: "Relativity, causality, FTL: pick two".
I choose causality and FTL. In a Newtonian universe, where there's a privileged reference frame, the speed of light isn't an absolute limit, and Einstein was wrong, what parts of physics would I need to re-work?
In particular,
Maxwell's equations, yes. Magnetism is closely tied to relativity. That's how the numbers work out anyway.
Two particles of equal charge will repel each other. But, if they move in parallel lines, there is an attractive magnetic force between them. Fun aside, when lightning strikes hollow objects, it shrivels them up because a very very strong current flows down the sides, causing an attractive force inwards.
The faster the two particles move in parallel, the stronger the magnetic force, while the electric force will remain constant. So, when do these two forces cancel? How fast do the particles have to travel in order to equal out the electric and magnetic forces? The answer, according to math, is C.
For a more clear picture, check out the two forces for electricity and magnetism:
$$F_{electric}=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r^2}, F_{magnetic}=\frac{\mu_0}{4\pi}\frac{qv}{r^2}$$
Nothing in any of those formulas is important except the constants. It turns out that
$$C=\frac{1}{\sqrt{\mu_0\epsilon_0}}$$
which is cool.
I don't know enough physics to explain exactly how they are related except for the above arguments, or how this affects things like permanent magnets. You'll no doubt have fun googling "why is the magnetic force is what it is" and turn up some cool stuff.
Anyways, from what I can tell, you might have to lose magnetism in your universe. That probably does it for nuclear reactions, too, considering it's called electromagnetic radiation.
Edit: As a disclaimer, I hope I haven't insulted your intelligence with this. You clearly knew they were related already or you wouldn't have mentioned those two specific points of interest. I hope this at least provides a decent starting point for showing how the magnetic force probably wouldn't exist in a universe without relativity, and, consequentially, light.
