Fallout from fission tests was understood fairly well by scientists prior to and definitely after the Trinity test. Fallout from megaton-range detonations required the Castle Bravo accident (1954) for a more complete understanding. Global fallout tracking was studied through the 1950s (e.g. as part of Project Sunshine), as were developed better models of fallout prediction.

Public awareness of fallout didn't really surge until after the Bravo accident, and even then fallout was not so much discussed broadly until the late 1950s and early 1960s.

I am not sure how one would measure public understanding of fallout — there are many subtle aspects that even elude experts today, much less the general public, in my present-day experience. By the 1960s though the popular representation of the effects of nuclear attacks talked about fallout, whereas the discussions in the 1940s and early 1950s mostly focused on just acute effects

Nuclear winter was actually fairly "born public": the initial paper on it by Sagan et al. was pre-scooped by a popular account by Sagan in Parade magazine in 1983. This was followed by the famous TTAPS paper in Science, the first real systematic publication on nuclear winter, and by many later further studies. This popular-before-peer-review approach was the source of some consternation by scientists, especially those who were already critical of Sagan in particular using this work to promote obvious political agendas. Lawrence Badash's A Nuclear Winter's Tale does a great job of describing this history and the later work on the subject, as well as the work that led to it on things like mass fire formation, ecological effects of nuclear weapons, etc.

Spencer Weart's The Rise of Nuclear Fear does a really good job of talking about the public vs. expert understandings of nuclear issues over time — it's the "one stop shop" for learning about a lot of this topic.