Not putting any probabilities on these ideas yet, just brainstorming:

I expect that some dynamics would be similar to nuclear strategy in the real world: nuclear weapons would make wars less common, but the few wars that did happen would have a risk of being much more devastating. If you're Saddam Hussein and you're pondering whether to start the Iran-Iraq war, maybe you hold back for fear of a nuclear exchange, and start your own local cold war instead. I'd expect that maybe we have only a small fraction as many wars, but some of the remaining wars would involve nuclear weapons.

While interstate wars might decrease dramatically, I'm not sure if the frequency of civil wars would change much. I could easily imagine a dark alternate history where maybe half of civil wars involve some small group (either rebels or an embattled remnant of government) getting their hands on a stray nuke and setting it off in the middle of a city.

It could be tempting for many countries to go for a North Korea strategy -- acting crazy and threatening everyone around them in exchange for concessions. Perhaps, after a bunch of initial chaos, the world would eventually enforce a much stronger norm against these rogue nations... imagine a kind of worldwide NATO who wouldn't have any qualms preemptively nuking dictators who seemed to be going down the North Korea path.

In a world where nuclear weapons were much easier to obtain and/or construct (imagine you don't need to refine the uranium at all, and you don't need specialized high-precision shaped explosives to set it off -- just take ordinary U-238 and wrap it in TNT), it would also become much more important to pay attention to other parts of a nation's nuclear capabilities. If you are a poor African country, do you have intercontinental missiles ready to fire towards anywhere on earth at a moment's notice, or do you just have some loose warheads that you can hope to sneak into another nation's port in a disguised container ship? Do you have any second-strike capability? Etc. Just giving everyone nukes would not put countries on an even, mutually-assured-destruction playing field. The resulting instability as different countries jockeyed to gain advantages over their neighbors -- more accurate missiles, stealthier submarines, etc -- would probably breed plenty of conflict.

Overall: A peaceful world stuck in a multipolar-Cold-War mexican standoff seems very unstable and unlikely to me. Instead, I think a world with easy nukes looks like something between the following two extremes:

1. Things are mostly similar to our world, except wars are a less common, but those wars often involve nuclear weapons, so everyone is just nuking each other occasionally, especially during civil wars, in poorer countries with less nuclear capability, and among dictatorships rather than democracies. (Cold-war-like dynamics mostly prevail among rich and powerful nations, just like IRL) This is obviously terrible because many more people die, and over time a larger part of the earth's territory, atmosphere, etc, is being irradiated.
2. In order to prevent continual low-level nuclear war as described above, the world coordinates much more to eliminate the threat posed by unstable and rogue nations. This coordination would take a different form depending on whatever part of nuclear conflict seems easiest to control. If our scenario is "nukes are still hard to make, but everyone gets 100 free warheads in 1950", maybe this means aggressively forcing countries to give up those nuclear weapons. If it's "nukes are easy to build, you just need the uranium", maybe the leading countries jointly occupy all uranium mines. If nukes are even easier to build than that, perhaps we end up with a neocolonial system with very few independent nations to eliminate the threat of war (no civil wars or unstable dictators if there are just a handful of empires dividing up the Earth), or a system of totalitarian surveillance if we are also worried about small non-state actors (as imagined would be necessary in Nick Bostrom's "Vulnerable World Hypothesis").

This seems like an interesting question! 

Here are some quick reactions (not really direct answers, and maybe not what I'd say if I'd researched this much more):

• I don't know of any sources addressing your specific question (though I imagine some might exist; I just haven't looked), but a good starting point would be sources on the more general topic of the ex ante historical likelihood of nuclear weapons strikes, the likelihood of future nuclear weapons strikes, and the factors pushing those things up or down.
  • On this, one of my top recommendations would be GCRI's A Model For The Probability Of Nuclear War
  • Another of my top recommendations would be How likely is a nuclear exchange between the US and Russia?
    • (I'm now working for the org that produced this post, and continuing this general research project, but I read this and thought it was useful a while before I got that job)
  • There are also some relevant questions on Metaculus, e.g. this one
• One factor that might reduce how many times they would've been used (compared to what a person would conclude if they ignored this factor) is that more proliferation might have reduced the overall number of wars/conflicts
  • See the nuclear peace theory
  • Though see also the "stability-instability paradox", a theory that "states that when two countries each have nuclear weapons, the probability of a direct war between them greatly decreases, but the probability of minor or indirect conflicts between them increases."
  • I still expect more proliferation would have increased the expected number of aggressive nuclear weapons uses over the period since 1950
    • But maybe less than one would think if one just imagined that the same conflicts would've occurred, but with more parties to those conflicts having nuclear weapons
  • (ETA: I've now seen that Jackson covered somewhat similar points, and I basically like that answer too.)

“[I know not what weapons will be used in world war 3, but I know that the next war after that will be fought with sticks and stones.]” My plain and immediate reaction in reading your question is “who knows, maybe we’ll survive one or two small-scale nuclear exchanges between regional powers, but civilization most likely will not survive to 2020.”

When literally every state has a nuclear arsenal, it doesn’t even take a crazed dictator to lead to nuclear war (although that is entirely possible), it just takes a few incompetent governments that

1. have faulty warning systems and risky nuclear postures and/or
2. can’t protect their arsenals from a committed terrorist group

To set off a chain reaction that quickly pulls more and more powers—eventually all major powers—into a nuclear exchange in a matter of hours (maybe days), not weeks/years like in other world wars. Eventually, the nuclear exchange will lead to nuclear winter, and civilization will collapse (which could lead to even more nuclear exchanges if there are still such capabilities). In the end, I think the number of catastrophe scenarios and the seemingly inherent instability of such a reality puts a really heavy burden on someone to disprove/rebut the idea that this would lead to destruction.