Approximately when does the story begin?
The story that seems to begin after an interesting divergent beginning (that goes into history about 10-20 years and almost 100 years ago...) sounds like it is 1932 or 1933. In the first line of the novel Scout tells us that this is the story about Jem being 13 and breaking his arm. We know that he's 13 by the end of the novel. By the end, Ewell looked for the job from the WPA (1935).
When the kids meet Charles Baker Harris (Dill), the introduce themselves and their ages. Scout's 6 or almost 6, and Jem is 10. The trial, is two summers later in which Atticus refers to 1935. This leads me to believe it starts exactly in the summer of 1933.
Only on one occasion in Harper Lee's novel, To Kill a Mockingbird, is a specific date mentioned. While Atticus is addressing the jury, he states "in this year of grace, 1935...," so we know the trial takes place during the summer of 1935. Scout states in the first chapter that "When I was almost six and Jem was nearly ten... That was the summer Dill came to us." The beginning of Part Two tells us that "Jem was twelve." This is the same year of the trial, so beginning with Dill's arrival, we can assume that the novel begins sometime in 1933 and ends on Halloween of 1935.
If you are asking in what year the story begins, all we can really say is that it begins sometime in the late 1930s. We can tell this because it is set during the Greate Depression and because the Nazis are in power in Germany. However, the war is not really going on yet so it can't be in the '40s. In fact, it can't be anytime after September of 1939 because that is when Germany invaded Poland, starting WWII.
We can also tell that it is not set in the early '30s because the WPA is already in existence and it was not established until 1935.