Arthur Flay was in the 26th Battalion the Royal Fusiliers - the Bankers' Battalion. The 26th was formed in 1915 and attracted bank clerks and accountants from all over the country.
Initial training took place at Marlow and High Beech. In November 1915, they moved to Aldershot.
On 4 May 1916, the 26th Battalion embarked for France.
Looking at CWGC and SDGW, his death was either on 15 September or 18 September, so I think he most likely lost his life during the Battle of Flers-Courcelette, which was part of the Somme offensive.
The week-long battle began on 15 September 1916.
http://www.cwgc.org/somme/content.as...ette&menu=main
The battle is remembered as the one where tanks were used for the first time.