So this is kind of a tricky question for a lot of reasons. Here's a stab at it.
First, let us consider the situation of chemical weapons (CW) and biological weapons (BW) after World War I. The use of CW in the war was noted by all, and it was seen as a particularly graphic example of how modern science could lead to weapons that were particularly cruel, unchivalrous, and undiscriminating in their victims. This in particular led to widespread condemnation. But we should be aware that similar condemnation was applied to some other weapons used during the war as well, notably the submarine and aerial bombing, and yet these were pursued with vigor in future wars. Similarly, the condemnation of CW managed to also slip and include BW even though such weapons had not been used and indeed many WMD scientists at the time considered BW to be quite different (and in many ways a "safer" weapon) than CW. What gives?
For one thing, one of the reasons that CW was deemed offensive but not "necessary" (unlike submarines) is that it wasn't actually all that decisive during the war. So that made it somewhat easier to marginalize. It is only difficult to condemn something if you actually like it on some level. When it came to trench warfare, gas just made things more unpleasant: masks for everyone, drifting onto civilians, but after the first attacks it lacked any significant strategic quality, because both sides started using it and developing countermeasures.
The slippage from CW to BW might be seen as a cultural one — the weapons have different sorts of characteristics from one another — but similarly BW suffered from the fact that its effectiveness seemed relatively low and unpredictable. Military generals prefer weapons that definitely work and whose effectiveness can be known beforehand, and weapons with less literal "blowback" potential (the wind blowing the gas back onto your own troops) than either CW or BW.
But anyway. Let us look briefly at World War II. Both Allies and Axis powers developed CW during the war, kept in stockpiles. Japan developed and used some CW and BW in China. US leaders did contemplate using CW in the Pacific (General Marshall suggested it might be the appropriate response to Japan's cave-fighting in the smaller Pacific islands), but otherwise but Allied and Axis leaders actively decided not to use CW or BW in their major operations against each other. Why? The answer here is somewhat murky — there are some indications of deterrence (Hitler didn't want to use CW because he thought it would be used in responses), some indications of cultural associations (Hitler had suffered from CW attacks and thought them uncivilized), and of course the Geneva Protocol of 1925, which explicitly banned CW and BW (the US, however, made clear that it saw this non-binding if the weapon was used against it first by another). How important was the latter? Potentially a lot, but there is quite a large debate amongst the International Studies people about whether treaties of this sort work because of their agreements or because they happen to establish norms that are convenient anyway.
We will note, however, that this was a ban on use, not manufacture or stockpile. And so this gets at one of the core contentions of the question: were the bans "successful," and if so, when? By many standards the Geneva Protocol was only successful in WWII in that it succeeded in limiting the use of the weapons (if indeed it was the reason for non-use), but not successful in that it did not make the use actually impossible (because it did not prohibit development of the technology, or its stockpiling).
The bans contemplated in the postwar for nuclear weapons (NW), like the Baruch Plan, were bans on development and stockpiling, not use. It was assumed that if the weapons existed, they could eventually be used — and that any pledge not to use them, even if backed up in a treaty, was hollow. It is an interesting cultural comparison with CW and BW in this respect, this difference in assumptions. It should be noted that many of the WMD scientists did not thing that NW should be lumped in with CW and BW — in the sense that they thought NW actually were weapons of mass destruction on an unparalleled scale, while CW and BW were not especially more uncivilized than, say, napalm or massed dropping of fragmentation bombs. Which is to say, today we tend to treat all three of these as WMD without controversy, but there were arguments at the time that there were real differences of scale between the three of them, with BW being the least destructive, CW being a little worse, and NW being an order of magnitude (or more) more terrifying.
(The attitude towards BW are in my view kind of fascinating because today we regard it with almost more horror than gas. I think this has more to do with our fears of modern biological application than we realize, though. The BW of the 1940s were focused on things like anthrax which, while a poison, is not some kind of super-plague that will kill everyone, the way we imagine BW would work today.)
OK, to wrap up. Many countries continued to have huge stockpiles of BW and CW through the Cold War. More countries developed BW and CW than did developed NW — they are just easier to develop. From a physical and industrial point of view, NW are just orders of magnitude harder to make and in principle easier to control the manufacture of (there are "hard" material constraints with NW; e.g., if you don't have access to uranium, you just can't make them).
The Biological Weapons Convention (1972) and the Chemical Weapons Convention (1993) were eventually passed. Let us look at the BWC — it was one of these treaties passed by Nixon (along with the Nuclear Non-proliferation Treaty) that sought to limit WMD production. The BWC is notorious today because it contained originally no real mechanisms for inspection or enforcement. It was basically a "pledge" of sorts. The US signed it willingly and in good faith. Why? Because they saw no future need on their part in BW. As Nixon put it to an aid, "If somebody uses germs on us, we'll nuke 'em." Which is to say, if BW were assumed to be part of the panoply of WMDs, then having more effective WMDs (like nukes) meant it was easy for the US to just forego them. (Get rid of nukes and maybe that doesn't hold, the logic might also go.)
The Soviets signed the BWC but regarded it as a hollow treaty. Some years later they started using new rDNA techniques to create new strains of plague and nasty viruses as a research program. They developed a capacity, should they choose to use it, to manufacture such next-level BW should they desire them, but they never produced them in bulk or integrated them into their doctrine as far as we know. (There's a whole separate story on that.) Later this all came out and these were destroyed. But it is just worth bringing up that it is not clear that the BWC was successful — it embodies for a lot of people exactly the failures of an imperfect WMD treaty, one that looks like it bans something but in reality preserves a pretty dangerous situation.
So. To conclude. "Why were bans on chemical and biological weapons more successful than attempts to ban nuclear weapons?" For one thing, it isn't clear they really were successful until relatively recently, and their success may have been in part based on the fact that CW and BW were simply not as effective as either weapons or deterrents as NW were assumed to be. This seems to me to be a large part of the answer — if you have limited funds for weaponry, you try to spend them on things that are reliable, might be usable, etc. The fact that there are negative stigmas attached to CW and BW makes them somewhat unappealing, but the fact that they aren't that useful probably drives them over that final cliff. The presence of NW may have in fact made it easier to ban them, too — by "raising the threshold" they made it easy for states like the USA to forego CW and BW.