There were some efforts made in this direction and much discussion of the topic. The UN Atomic Energy Commission was created in 1946 with the express goal of exploring whether "international control" — aka a workable ban — could be implemented. The US proposal, the Baruch Plan, set out a system by which this might work, but it allowed the US to keep its existing arsenal until the USSR submitted to international inspection. The USSR would not consent to this. The Soviet proposal, the Gromyko Plan, basically said that no nuclear weapons would be made or used, and that all existing nuclear weapons (in the US stockpile) would need to be destroyed within 3 months. It had no verification measures, however, so the USA definitely would not support it (they did not trust the Soviets).

On these disagreements and distrusts, the idea was effectively dead in the water by late 1947. We now know that the US was somewhat wary in general of international control (it continued to develop and even test weapons during this period) and the Soviets were rushing as fast as they could towards their own nuclear capability (and saw acquiring such weapons as absolutely essential to their own security), so the odds of such a plan having worked appears to be pretty low no matter how you split it. The US and USSR would not really get serious about curbing weapons development until the 1960s-1970s, when they agreed to a number of treaties meant to slow the arms race, limit deployments, and curb nuclear weapons development by new nations (non-proliferation).

Separately: chemical and biological weapons were not in a meaningful way banned after WWI. The Biological Weapons Convention was drafted in 1972, and the Chemical Weapons Convention was drafted only in 1992. The Geneva Protocol of 1925 does ban the use of such weapons but not their production or stockpiling, and some key countries (e.g. the United States) did not even ratify that part of the protocol until much later. Many ratified with the reservation that they would use such weapons if an enemy violated them, which is a weak treaty indeed.