Edmund Busche was killed on the Western Front in WW1, on May 2, 1916. This was reported by his colleague Paul Riebesell in the annual report of the German Mathematical Society:
Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 25, 1917, p. 283 (Google snippet)
E. Busche †
Von P. Riebesell in Hamburg
Am 2. Mai 1916 fiel als Hauptmann der Landwehr a.D. an der Westfront Edmund Busche im Alter von 55 Jahren. Der Verstorbene war seit 1888 Mitglied der Mathematischen Gesellschaft in Hamburg, seit 1891 auch Mitglied der Deutschen Mathematiker-Vereinigung. Nachdem er etwa 20 Jahre als Oberlehrer an der Hansaschule in Bergedorf tätig gewesen war, wirkte er in den letzten zehn Jahren als Professor an der Oberrealschule in Hamburg-Eimsbüttel und als Dozent [...]
From this we learn that at the time of death he held the rank of captain in the militia. Edmund Busche had been a member of the Hamburg Mathematical Society since 1888 and a member of the German Mathematical Society (founded in 1890 by Georg Cantor) since 1891. He had worked for about twenty years as a senior teacher at a high-school in the Hamburg suburb of Bergedorf (now a borough within the City of Hamburg), and thereafter for ten years as a professor at a high-school in Hamburg-Eimsbüttel.
The Hamburg Mathematical Society honored Edmund Busche in volume 5, issue 6 of their journal Mitteilungen der Mathematischen Gesellschaft in Hamburg. A high-quality scan of the issue with a picture of Busche is provided by the HathiTrust Digital Library.
The issue starts out with an article by E. Hoppe in Hamburg "Edmund Busche zum Gedächtnis" (In Memory of Edmund Busche) which provides a biographical sketch of Busche's life. He was born on May 2, 1861 in Neuland in the district of Kehdingen which is on the Elbe river north of Hamburg. He was the second son of a minor official. Soon after his birth the family moved to the small town of Drochtersen where his father died in 1868. His mother then moved the family to Walsrode, where they lived for three years, before moving to Uelzen in 1871. In 1875 they moved to Hanover where Busche completed high-school.
He then attended the University of Göttingen to study mathematics. After only four years he completed his dissertation under Ernst Schering, finishing his doctoral studies on February 27, 1883. A few days later he passed the teaching exam, allowing him to work as "Oberlehrer" (senior teacher). He then served in the military, where he ultimately achieved the rank of lieutenant. Due to a glut of teachers he initially had trouble finding work and had to work as a private tutor for a while before finding permanent employment with the Hansaschule in Bergedorf in 1886.
He got married in 1901 and had two children, a girl and a boy. According to genealogical information I found on the internet for whose accuracy I cannot vouch, his wife Ottilie Fastenau was born in 1876, his daughter Friederike in 1902 and the son Reinhard in 1904.
The next article in the commemorative issue is by H. v. Mangoldt in Danzig, "Edmund Busches wissenschaftliches Lebenswerk" (The Scientific Œuvre of Edmund Busche). Including his Ph.D thesis, it lists thirty-five publications by Busche. The author points out that it is clear from the topics of the publications that Busche's interests lay entirely within pure mathematics, with a focus on number theory and projective geometry. He mentions that since Busche gave a talk on the theory of relativity at the Hamburg Mathematical Society in 1911 he must have also had a passing interest in mathematical physics.
The next article is by P. Riebesell in Hamburg, "Eine Verallgemeinerung des Pascalschen Satzes für beliebige Sechsecke nach E. Busche" (A Generalization of Pascal's Theorem for Arbitrary Hexagons, based on work by E. Busche), in which the author attempts to complete Busche's final, unfinished, work. In the introductory paragraph he mentions that Busche served on the front in Belgium. He last saw him when he was furloughed from the front in January of 1916, at which time Busche asked him to do some background research on this topic.
Busche's 1883 Ph.D. thesis is entitled: "Ueber eine Beweismethode in der Zahlentheorie und einige Anwendungen derselben, insbesondere auf das Reciprocitätsgesetz in der Theorie der quadratischen Reste" (On a method of proof in number theory and some applications of the same, in particular to the reciprocity law in the theory of quadratic residues) and is dedicated to his mother. A high-quality scan is available from the Göttingen Digitization Center. At the end of the dissertation there is a brief biographical sketch that confirms information from other sources: Busche was born on May 2, 1861 in Neuland in the district of Kehdingen. His mother's name was Friederike. In 1879 he finished high-school in Hanover. From Easter 1879 through Easter 1883 he studied mathematics in Göttingen. On March 3, 1883 he passed the teaching exam and subsequently departed for mandatory military service to Berlin. He thanks his Ph.D. advisor, professor E. Schering, for much valuable advice.
An annual report of the Hansaschule from 1889 and a festschrift published by the school in 1908 confirm some of the biographical details already mentioned above.