I have unearthed a history of the School that shows that the School OTC wore the RE cap badge from its association with the West Yorkshire Engineer Volunteers during the Cadet Corps era.
'. Most school corps wore the cap badge of their local infantry regiment to which they were attached, but the KES contingent continued their connection with the Royal Engineers and wore their badge instead, although after the war they
changed to wearing the school’s badge on their cap.'
A photo of a KES cadet with the RE badge is here:
http://oldedwardians.org.uk/nlc/arti...ingTedsJCC.pdf
The OTC cap badge shown in K&K was introduced post war and can only have had a short life as the OTC was disbanded in 1927 by the Sheffield Labour council. The school raised an ATC in 1941.