After a brief spell in 1940 when a single NZ brigade was deployed in Scotland, I am not aware of any NZ units that trained in the UK. No NZ land units either landed in Normandy or fought in France and the Low Countries. My understanding is that from 1940 onwards NZ troops went to the Mediterranean and Aegean theatres via East Africa or India, and then Egypt, but happy to be corrected.
As for grenade supply, 1944-45 saw orders placed by UK Govt to NZ for 3 million No.36. These were for supply to Burma, Egypt (for Italy) and UK, which explains the presence of NZ base plugs on UK grenade ranges.
Slightly off topic, for supply of Canadian-made No.36, the Department of Munitions and Supply (DMS) monthly shipment records show Canadian factories shipped:
July 1944: 425,000 to UK Govt, and 20,000 to Canada (training);
August 1944, 1 million to UK Govt, and 30,000 to Canada (training);
September 1944, just over 1 million to UK Govt, and 30,000 to Canada (training).
A similar story for other land service stores, including No.69, No.75 and No.77 grenades, and 2-inch and 3-inch mortar bombs - small numbers for training troops in Canada, much larger quantities to UK depots. All troops fighting under Montgomery's 21st Army Group ( British 2nd Army and 1st Canadian Army), in Europe were issued with grenades that came from those depots, and they could have been made in the UK, South Africa, NZ or Canada.