The word seems to have been first applied shortly after 1910 to the kind of music that had arisen in New Orleans; and it wasn't long before we had bands calling themselves 'Jazz Bands'. By the way, it was often spelt jass in those early days.
The Original Dixieland Jass Band was making records by 1917.
The word and the music style probably became known in Europe the same year.
The first so-far documented use of the term in my country - England - was a reference in July 1918 to a concert by The Liberty Jazz Band for 835th American Aero Squadron at their base in England near the end of the First World War. Presumably this was an all-American band, possibly drawn from members of the Squadron's military band who were trying to imitate The Original Dixieland Jazz Band.