That's pretty much were I am right now: trying to figure out how much should work differently and how much can be hand waved or swept under the rug. I can deal with simple things like magnetism not existing, having no universal speed limit, mercury being a solid at room temperature and so on, but I know that pretty much everything should be different. Maybe it's possible to just say that atoms and chemistry happen to work very similarly to our universe, but it's pretty unsatisfying to say that without at least considering as many aspects as I can.
Starting with atoms, for that matter, would covalent bonds not exist? I actually don't know. I strongly suspect the orbitals would be different at the very least, which impacts chemistry in profound ways, but for all I know relativity is fundamentally important to the idea of electrons even entering into orbitals instead of just colliding with nuclei and turning the universe into a soup of boring neutrons. And yet, would neutrons even exist? Again, for all I know, the strong force, quarks and so on are entirely different without relativity... I've heard that time dilation has important consequences on the decay rate of what I believe were charge carrying particles, which impacts the strength and range of those force fields, but this is all have remembered details from something I read a long time ago. Can't remember if it was the strong force or weak force, or something else entirely.
Hmm... that gives me thoughts to file away for future consideration - what would a universe without particle decay be like? Unrecognizable, I'm sure. Or a deterministic universe without uncertainty? Again, probably unrecognizable.