By July 1916 the training base for the whole of the New Zealand Expeditionary Force had been transferred from Egypt to Sling Camp, Salisbury Plain, this being more suitable for troops destined for service in France. Subsequently, in Semtember 1917, the New Zealand Rifle Brigade training base was transferred to Brocton Camp, Cannock Chase. This is still the site of a memorial to the Brigade mascot "Freda".
http://www.nzhistory....-mascot-freda-the-dog
As far as I can make out it would be far from normal (if at all possible) for a British subject resident in Britain to enlist in a NZ regiment, even one based in Britain. Is there any chance your great uncle could have found his way to NZ before enlistment?