It is extremely difficult to avert the nuclear arms race with a PoD in 1945. The arms race was the product of a complex interaction of national security desires (in both the United States and the Soviet Union) and institutional incentives (again, in both countries) that your proposal does basically nothing to deal with.
As far as national security goes, the questions at hand were rather simple, on both sides. To begin with, it was a question of how do I beat the Soviet Union/United States if we end up fighting? On the part of the United States, it was plainly obvious at the end of the war that the Soviets had a much stronger conventional army than the United States could feasibly support (or wanted to support), and that they could relatively easily conquer Western Europe through force of conventional arms if they so desired--and given Stalin's mercurial nature and previous aggressive actions in Eastern Europe (not to mention his forthcoming aggressive actions in Eastern Europe, and the anti-communism of U.S. leadership) it could not be ruled out that they would so desire. This would be extremely bad from the position of U.S. national security, since the Soviets would then essentially be in the position of Nazis if they had completely defeated the Soviets during Operation Barbarossa; they would completely control the European subcontinent aside from the British Isles, and would therefore have full access to all of its resources to build up navies and air forces capable of invading Britain or, potentially, the United States, while the United States would have much lesser resources available to meet them.
Thus, something other than conventional arms needed to be found to prevent this nightmare scenario from occurring, something that could plausibly defeat the Soviets if they decided to invade and would thus deter them from being aggressive. The obvious answer, even if Hiroshima and Nagasaki are not destroyed, is nuclear weapons. Why? Because they are quite clearly much more powerful than any conventional weapon, and have the potential to destroy entire cities or military units. This was more than adequately demonstrated by the Trinity test and subsequent nuclear tests, which would surely take place anyway once the Manhattan Project advanced to the stage it was at by the beginning of 1945, and even basic calculations of the amount of energy that could be liberated by fission would show that a nuclear bomb could be a city-killer. Moreover, building up a large nuclear force strongly leverages the strengths of the United States in advanced technology and a large industrial base, while downplaying relative weaknesses in population and the need to transport armies across oceans and supply them there, and offered an attractive way to potentially achieve national security objectives without the expense of a large army or navy.
On the Soviet side, the incentives were very similar. While the Soviets had great strength of arms in 1945, they could not guarantee that in the future the Western powers would not build up a stronger army with the overall greater resources and population available to them, and then take the route of Barbarossa and attack Russia. This had, after all, happened twice in the past forty years (indeed thrice, if you count the Civil War interventions separately from the German invasions), and previously in 1812 France had invaded Russia, while all of the principal Western leaders were rather anti-Communist. Additionally, the Manhattan Project was well advanced and the Soviets knew a great deal about it due to their intelligence penetration of the operation, including how powerful a nuclear weapon would be. Clearly, having many nuclear weapons and being able to deploy them onto Western cities or armies would be very beneficial from the point of view of a Russia facing Western invasion, as well as preventing any form of nuclear blackmail from the United States that might take place if the United States believed it was the only power possessing nuclear weapons. In fact, Stalin had already begun a nuclear program by 1943, albeit at an understandably low priority under the circumstances.
In other words, from the point of view of both sides there were enormous incentives to build up nuclear stockpiles in order to ensure their own security against aggression by the other. Fundamentally, the issue is that the West has no reason to trust Stalin (or the Soviets more generally) to keep his words after things like the Molotov-Ribbentrop Pact or the Communist takeover of Czechoslovakia. In reverse, Stalin has no reason to trust the West after Barbarossa and the Western (French and British) abandonment of Eastern Europe prior to World War II. So both sides have reasons to be very suspicious of the other, and reason to want to ensure their own national security through a large-scale nuclear build up. Hiroshima and Nagasaki only offered a final proof that nuclear weapons could devastate cities in a single blow, which was not truly necessary to convince leaders that they could be a very powerful weapon. You might argue, however, that "only a few hundred weapons" would be enough to ensure deterrence. That is...not entirely true when it comes to countries like the United States and Soviet Union. Because of the sheer size of each country and the large number of cities and other possible targets within each country, it's actually quite easy to write up target lists with many hundreds or even thousands of targets that might be worth nuking. Factor in that nuclear weapons and their delivery systems are not perfectly reliable--some, indeed, have been almost totally unreliable--and you quickly find that you need a huge number of weapons to ensure destruction of all of these targets. Obviously the other nuclear powers have gotten away with much smaller arsenals even so, but this was more a matter of available funds rather than strategic logic--the British, French, and Chinese simply could not afford strategic forces large enough to destroy everything that might be worth destroying in the enemy country, so they had to get away with the bare minimum of what they thought would be intolerable to their enemies.
This is where we get into institutional incentives, in this case armed forces incentives. On both sides, dedicated branches of the armed forces became responsible for nuclear weapons, particularly strategic nuclear weapons--in the United States it was primarily the Strategic Air Command of the Air Force and to a lesser extent the Navy's missile submarines, while in the Soviet Union it was initially the Air Force and later became the Strategic Rocket Troops. Like any armed service, these had institutional incentives for wanting more nuclear weapons--this meant more money for them, more personnel, more bases, basically everything that a general could want in terms of status and power. So, again on both sides, there was a big incentive to hype up the threat posed by the other side and argue that more nuclear weapons were needed to ensure security. This was only compounded on the U.S. side by the fact that many of the leaders of the Air Force until the mid to late 1960s were bomber men, and more nuclear weapons particularly bolstered them (it is probably no coincidence that the U.S. stockpile peaked in the early 1960s), and by the fact that a U.S. desire to minimize conventional arms spending and disarm so far as possible created gargantuan incentives on the part of all three services to try to get part of "the nuclear mission" and the funding that went along with it. Things such as the Army trying to get into strategic missiles and the Navy trying to build the USS United States and later the Polaris-class SSBNs should be seen in this light, and also played a significant role in the U.S. arsenal growing so much up to the early 1960s--until then, it was simply much easier to get funding if you could claim a nuclear mission. I am less certain of the situation in the Soviet Union, but I expect that similar circumstances obtained. Certainly the Strategic Missile Troops would have had a large incentive to push for more nuclear weapons, since they were literally the only thing they did, and probably the other armed services would have also had significant institutional incentives to expand the arsenal, as in the United States, even though the Soviets were never as enamored with the idea of the all-nuclear battlefield as the United States (fascination with wunderwaffe is an especially American phenomenon).
So, how do your proposals integrate with these facts? Well, I've already discussed the absence of the nuclear bombings of Hiroshima and Nagasaki, but to reiterate these not occurring will have no discernible effect on the degree to which Stalin and the United States want nuclear weapons. As I explained above, even without them being used they had been and were going to be tested, and this and some basic calculations would show that they had city-destroying capabilities. Therefore, both sides have great incentive to want them regardless of whether or not they have been used. Having a demilitarized and neutral Germany will have a somewhat greater effect, but not a large one. It reduces the degree to which each side will fear a "bolt from the blue" surprise conventional attack, but such a Germany is, well, weak (by design) and could be easily invaded and crossed by the other side. Therefore, both sides will still want powerful conventional and nuclear forces in reserve just in case the other side decides to ignore German neutrality and invade across it, which gets right back to the start, especially since armed forces incentives would then make them want to increase nuclear arsenals still further "just in case." You might decrease the incentives for tactical nuclear weapons, particularly on the Western side (which otherwise had zero hope of stopping Soviet forces, especially on the Inner German Border), but you're not going to do anything for strategic weapons.
A somewhat more promising angle to take would be the Acheson-Lilienthal or Baruch Plans for nuclear disarmament proposed by the United States in 1946. This would have involved a U.N. department taking control of nuclear energy and nuclear technology globally, in particular uranium mines, and ultimately becoming custodian of all nuclear weapons. As with the Stalin Note, however, this foundered on the fact that the Soviets and the West did not trust each other and both wanted the other side to do something that the other side did not want to do (disarm, on the part of the Soviets vis-a-vis the United States, and submit to inspections on the part of the United States vis-a-vis the Soviets) before doing something that the other side did want to have happen. This one is a bit easier to resolve, because the U.S. acceding to the Soviet proposal would actually have very little effect on U.S. national security, due to the very small size of the arsenal and the few delivery systems available at that point in time. Nothing in the initial disarmament would prevent the United States from maintaining the infrastructure needed to build more nuclear weapons if a final agreement proved impossible to form, nor from building more weapons down the road if it proved necessary.
In fact, this was the fatal flaw with the plan, and pretty much all disarmament plans proposed at the time. See, in order for the plans to be effective, there needed to be some mechanism of control to ensure that countries did not agree to disarm and then build more weapons anyway (see what happened after the Biological Weapons Convention, when the Soviets invested a lot more in biological weapons out of the belief that the U.S. proposal of a treaty with no verification mechanisms simply meant that they were trying to pretend that they weren't running a biological weapons program). The method of control used by these early plans was to control uranium and thorium mines and monitor the amount of material mined, under the belief that those materials were rare and there were not so many deposits of them. The problem is that belief is completely wrong. In fact, there is loads and loads of uranium and thorium, and it would be very difficult to control their mining. A more promising route is the NPT approach of controlling enrichment technology, but this also has its limitations, as shown by the relative successes of the Iranian, North Korean, Pakistani, and Indian nuclear programs against NPT controls. Since most nuclear reactors use enrichment to some degree, it is very difficult to permit civilian use of nuclear technology, which everyone agreed and still agrees should be possible (if nothing else, a lot of useful medical isotopes are produced at reactors) without thereby opening avenues for diversion of material to military use. Of course, this would still result in much smaller arsenals, at least until countries feel empowered to openly break with the treaty regimes.
Still, if you really insist on a 1945 PoD, somehow getting the United States to surrender nuclear weapons first in something like the Acheson-Lilienthal or Baruch Plans would probably be your best bet, certainly the earliest reasonable PoD. This is admittedly very difficult, despite the logic I laid out earlier, because of the critical role of the U.S. Senate in ratifying treaties (and without ratification a treaty is not operative). The problem is that under the Constitution a 2/3rds majority is needed to ratify a treaty, and you would almost certainly find more than 1/3rd of the Senate in the 1940s unwilling to ratify an arms control treaty with the Soviets that would see the United States giving up its nuclear arsenal regardless of the sense or utility in doing so, due to the large number of anti-Communists in the Senate. Although the United States could abide by the treaty anyway through executive action, as it has done with the CTBT despite not ratifying it, this situation is not stable and would almost certainly result in the Soviets ignoring the treaty and developing their own bombs anyway, since they couldn't trust that the United States would continue to abide by it when the administration changed. Thus, it probably wouldn't work...but it's more likely than just about anything else.
www.hup.harvard.edu
en.wikipedia.org
www.wilsoncenter.org
So I don't take this as definitive proof that my proposed scenario is implausible.
