Back on page 8 of this thread, my sister posted a medals document with a few names on it, including KEELING, William E., our grandfather.
At the top of the document, two places above our grandfather, is a Sgt. THOMPSON, William E.
We have several photos of a Sgt. Ted Thompson, in Leighton Buzzard, and some locations in France (specifically, Bailleul and Yvetote).
I guess that "E" stands for Edward and it is the same man listed in the medals document.
Here is Ted Thompson, in Leighton Buzzard, July 1915. I think he is not yet a sergeant here: