Born in Berlin, after attending the Oberrealschule and subsequent years as a commercial apprentice, Bumcke took lessons in composition with Gustav Kulenkampff (1848-1921), Max Bruch and Engelbert Humperdinck, piano with Hugo Rüdel and Otto Neitzel, and trumpet with Julius Koslek (1835-1905).
For lack of good saxophonists, Bumcke played many of his compositions himself and in 1927 founded his own saxophone class at Stern's Conservatory, then later at the Klindworth-Scharwenka Conservatory, from which especially Bumcke's daughter Hilde became famous as a German saxophonist under her stage name Ingrid Larssen.
In 1933, he reconciled the National Socialist cultural politicians with the instrument saxophone, which could continue to be used in German dance orchestras.
[2] Together with Sigurd Rascher, Bumcke is considered by experts to be a pioneer of the classical saxophone in Germany.
From 1950 to 1955, Bumcke was a lecturer in music theory at the Hochschule für Musik "Hanns Eisler" in East Berlin.