While it may have been known that a need for safe storage of nuclear waste existed before 1943-1944, it was certainly known by 1944.
The Los Alamos National Laboratory began research and development on the first nuclear weapon in 1943 LANL under the Manhattan Project which lasted from 1942 to 1946. The B Reactor at the Hanford site in Washington state was the first reactor to produce plutonium on a full scale, and that was the source of the first nuclear test bombs, as well as the actual bombs dropped on Japan. That reactor was built and in production by 1944.
The experiments, and this development, created a substantial amount of waste. This is the sort of waste that once created in and around 1943-1944 would immediately present itself as being quite hazardous and thus in immediate need of safe disposal.
While I do not see any sources telling us that droves of scientists knew that this material was toxic, nor do I see that droves of scientists had to die before we discovered that this material needed to be stored safely, we can deduce that certainly by 1943 there was a need for specialized waste disposal and storage by virtue of the fact that they were quite careful to store experimental waste underground in special containers at this time. Many of these early containers are causing trouble today, and that is old news.
At Hanford, waste was stored in underground tanks, and these are leaking today. I quote and source, "Some underground storage tanks filled with radioactive sludge from as early as the 1940s are slowly giving out, with 67 known to be leaking into the soil.". The fact that they tried to store it implies they knew of the problem.
This source suggests in the first sentence that "officially-designated material disposal areas" started at Los Alamos in 1944.
Generally it does not take days for scientists to recognize this sort of waste as being quite toxic and naturally the need for waste storage would occur as soon as the waste starts being created. While I say that the answer to your question is certainly by 1943, it appears that the Nazis buried what appears to be nuclear waste, and this has been a modern surprise, as historically no one thought the Germans had a program that was advanced enough to produce nuclear waste. Source for that: Nazi nuclear waste.
I am convinced that certainly by 1943 the need was known, and your question asks for a date. I suspect however that, in light of the human knowledge gained since the time of Marie Curie, that the scientists whether German or American knew quite well that they were about to create hazardous waste material that would need to be stored safely before they even began doing their work. In that light, someone may have known sooner than the WW2 start of nuclear weapons development that the process (theoretical before then) would require waste storage.