Until 14 November Ancestry.co.uk have free access to their military records. Not one to look a gift horse in the mouth I've taken advantage of this. A search for Gunnee returns 8 results: George Gunnee (the same record twice) with his army service records and Alfred C Gunnee, Alfred Gunnee, S Gunnee, Samuel Gunnee, Tom H Gunnee from the 1914-192 Medal Rolls and W Gunnee from the UK, Naval Medal and Award Rolls, 1793-1972 (not included in the free access).
Sadly the record for Alfred Gunnee is actually for an Alfred Gunnell and Alfred C Gunnee is Alfred C Gunnell. S Gunnee is actually S Gunnel.
George Gunnee's (born 1892) record is for his service in the RASC - he signed up on 1 October 1914. By April 1915 he was promoted to Sergeant. he was 'disembarked' in September 1919.
Samuel Gunnee's record is genuine and is the record of his Victory Medal for his service in the Royal Army Service Corps.
Tom H Gunnee is also genuine showing service in the East Yorks R, and then the RAF.
So now I have more research to do!