If its about cold war levels of MAD then obviously war between the major powers would be deterred. Deterrence worked during the cold war so why wouldn't it work during the 1930s if similar weapons were around.
Contrary to some popular belief, ww2 didn't happen because people went insane for a while and then became normal again after 1945. Many aspects of nazism and fascism were declared to be insane after 1945 but the events of our current time show, that in fact these things can repeat and people may more or less willingly, pushed by circumstance, go down the same route again and dive major nations again into fascism.
Reversely if nukes were around in 1939 then a lot of the thinking of the cold war would be invented rather quickly to cope with the problems of deterrence. Nazi Germany will for obvious reasons not attack France as directly at they did in 1940 if this is likely to invite MAD the moment the Germans break through towards Paris. And Germany will not attack Britain with aircraft or missiles if this puts Britain into a situation where such attacks cannot be distinguished from a nuclear first strike, thus inviting MAD on Germany.
Poland though, unless the British and French put serious trip wire forces there, will get crushed as in OTL. As Russia wished to crush Ukraine. France and Britain could theoretically declare that they would invoke MAD but that's very hard brinkmanship, too hard for the domestic political tolerance of such things in both France and in Germany.
In fact if it looks as if Germany was to declare war on Poland, France and Britain might well not ally with Poland as they did in OTL. They might consider the risks inacceptable just like nato in 2022 found the risks associated with no fly zones or things like that unacceptable in Ukraine.
But they might openly ally Romania, move troops and aircraft and even nuclear weapons into Romania, and thereby draw a line there. Romania doesn't border Germany so it's a place where they can draw a line and still have some room for preventive measures and escalation control. And because Romania (pee 1940) has a border with Poland, they can help the polish army escape, provide bases to them in Romania, and keep up a guerilla war in Poland, using Poland as a proxy against Germany.
Stalin would not move against Romania if Romania were covered by a solid Franco British guarantee and military bases there.
A direct German conventional attack on France is made unlikely by the nuclear deterrence. Almost impossible. Almost. But if you think about how close the distances in Europe are, how short the flight time of missiles, and how little strategic depth even a country like (metropolitan) France has if viewed through the lens of nuclear strategy then it's not entirely impossible for Germany to dare an extremely risky move and strike fast, and hard, into France. The idea being (similar to some thinking about west Germany during the cold war) that the French were suddenly and unexpectedly put into a situation where German troops were already deep in French territory, they would "blink" and hesitate to nuke themselves.
Kind of difficult to envision though. Because the moment Poland gets crushed by Germany, Belgium will definitely get much worse panic about their security than in OTL and choose to seek a fast and direct alliance with France, inviting French troops and nukes onto their soil. And then the Ardennes strike idea becomes more even risky for Germany so they might not dare it.
Thinking about France again - France has colonies in Africa. This does give them some strategic depth in that they can put ICBMs into Algeria and Tunisia which can glass Germany but which Germany cannot touch with conventional weapons. They would definitely do this, I think. They would also do the kind of doomsday planning that the otl cold war powers did, war gaming nuclear Armageddon over and over, and figure out that for political reasons they can't move the capital away from Paris but they definitely can prep ahead of time for a huge government bunker complex to be built in the mountains south of Algiers. If conflict looks likely to erupt in Germany, they'll move the vice premier and parts of the government over to Algiers as a backup.
Germany has no such luxury, their strategic depth is almost nonexistent. Missile flight time to any point in the greater German Reich is short. Once they gobble up Poland and border the USSR, they're going to be quite paranoid about having a backup. Historically during WW2 they moved a lot of industry into Thuringia, Bohemia and Silesia as those places were more distant from the British and American air forces than the Ruhr, Stuttgart or Berlin.
Given the constellation of powers and with nuclear deterrence in play, I could see Germany drifting into the position of a Soviet client over time? Like today's Russia to today's China. Purely materially they are decent partners for each other - Germany has a very highly developed industrial and military technology, the USSR has very rich resources. The USSR with nukes, and with the extremely low hesitancy to destroy parts of their territory, will be impossible for Germany or anyone else to directly attack without near certain destruction. Definitely no Barbarossa.
The Baltic countries are a goner, as is Finland. Bolshevism it is for them.
Scandinavia - proxy war?? Germany might try to invade Denmark but not Norway (too easy for Britain to cover them, so the risk won't be worth it). But Sweden, that's doable. So, Sweden might learn the joys of national socialism. Norway then might become theater of a proxy war. Or not, if they keep a tight lid on domestic security and preemptively arrest all the nazis and communists.
Italy with nukes will want to conquer Greece like in OTL. Mussolini does a Putin: "Anyone-a who meddles with us-a, is-a gonna see-a conseguenze the like of which-a you have not-ah seen!" And it might go over as well as it did for Putin, with the expected quick victory not materializing (as per OTL). He'll be pushed back to his border as French and British supplies to Greece empower them even more than IOTL. After all, with conventional deterrence in France being slightly less critical due to nukes, France and Britain can afford to spare with more stuff to Greece (on top of what they give to the Polish underground via Romania)
The world might settle into a confrontation with USSR and Germany forming, by necessity, an unholy Eurasian alliance, and Britain/France standing on the defense across Europe. Without direct attacks on each other, the full military might of birth sides will be funneled into proxy wars in Poland, Sweden, Greece, and then other places too. Basically everyone who's not a nuclear power, and not credibly protected by one, will suffer nightmare levels of destruction. Romania might actually avoid it due to their unique position. Yugoslavia is a goner, though. Carved up between Germany and Italy and then turned into a gigantic proxy battle ground. (Sounds eerily like OTL though...??)
The overseas colonies of France and Britain might become battlegrounds as the USSR will see it in their direct interest, there as IOTL, to push decolonization and spread weapons far and wide.
As a cold war constellation it actually looks very bad for France and Britain! USSR plus Germany is a formidable monster of an alliance, and without the USA on their side the two are going to face a difficult uphill struggle defending their positions. Decolonization is almost a certainty to happen sooner than IOTL because of the pressure.
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