My question is, What was the exact date when Prosper and Bo ran away from Hamburg, Germany?
We learn quite a bit about the setting of Cornelia Funke's young reader's novel The Thief Lord in the very first chapter, which helps us figure out the approximate time the children ran away from Hamburg.
In the very first sentence of the book, we learn that it is "autumn in Venice." We also know that it is far enough into autumn that winter feels like it is fast approaching and that "even the air in the alleyways tasted of snow." From these two points, it's safe to suppose that the novel is set in either late October or early November. When Mr. and Mrs. Hartlieb approach Victor the detective, we also learn from their conversation that their nephews Bo and Prosper "ran away more than eight weeks ago" (p. 10). If we suppose the novel is set during early November, we can conclude that the children have been missing since some time early September. Though we can't state an actual date, we can approximate that they ran away in September.