When was madame butterfly set?

Madame Butterfly was first performed in 1904, and the story-- at the time-- was supposed to be contemporary. So, the safest answer is that it takes place in Nagasaki, Japan, at the turn of the 20th century.

But, if we go by the "real Madame Butterfly," on whose life the original short story and libretto are based, then the events would be set in 1870.