The Boy Sherlock Holmes, His First Case: The Eye of the Crow by Shane Peacock

From chapters.ca:
Sherlock Holmes, just thirteen, is a misfit. His highborn mother is the daughter of an aristocratic family, his father a poor Jew. Their marriage flouts tradition and makes them social pariahs in the London of the 1860s; and their son, Sherlock, bears the burden of their rebellion. Friendless, bullied at school, he belongs nowhere and has only his wits to help him make his way. But what wits they are! His keen powers of observation are already apparent, though he is still a boy. He loves to amuse himself by constructing histories from the smallest detail for everyone he meets. Partly for fun, he focuses his attention on a sensational murder to see if he can solve it. But his game turns deadly serious when he finds himself the accused — and in London, they hang boys of thirteen.