What year was the Buffet Crampton clarinet with serial number 8478 manufactured?

Buffet Crampton did not manufacture any clarinets under that serial number. They started their serialization from 12345 in the early 1930s. However, Evette & Schaeffer, which later merged with Buffet Crampton began its serialization from around 3000 in 1908. It is possible that your instrument is a clarinet made by Evette & Schaeffer, but there is no way to determine its manufacturing date without more information.