The only listing around is a census of vehicles done in 1949.
This is now held by the Army museum in Waiouru.
The list shows NZ numbers and a description of the vehicle make and model, country of origin, and which of the 3 military districts it was in at that time (1949). At this time the standard nomenclature used overseas was adopted, so there are some repeats where a vehicle is shown by new and old descriptions.
There are very few early NZ numbers in the list as these seem to have been issued to impressed vehicles and the existing prewar vehicles, which were all disposed of during the war or soon afterwards.
There is no pattern to the numbers, except for some large batches of vehicles like Indian Motorcycles and GMC trucks with sequential numbers. Most of the other vehicle types are scattered with small numbers here and there.
My guess is that they were issued perhaps at depots or at the units as they received them. I don't think they were issued at the assembly plants.
Period photos are the best clue, but they are not plentiful. Often the markings are visible but not the NZ number, and vice versa.
As far as overseas service goes, I believe NZ based vehicles were used solely in the Pacific Campaign (Solomons), by the south pacific island garrisons New Caledonia, Fiji, Tonga etc, or for training at home.
There is a reference to some trucks being sent from NZ to the Middle East, (which may have been Ford 4x2 types), but remember NZ was in dire straits in 1942, with the Japanese advancing south.
The LP2 carriers were built between late 1941 and 1942, and the orders for tactical vehicles (tracked armour and all wheel drive soft skins) not arriving till late 42 and into 43.
El Alamein was Oct 42, and there had been a flood of vehicles arriving in North Africa to reinforce the 8th Army before then. (Some obsolete vehicles like CMP 2 pounder portees came to NZ (and Aussie) as the guns were still effective against Japanese armour).
Shipping was at a premium at this time. From early in the war, NZ was supplying food to the UK, (and later to US forces in the South Pacific), as well as reinforcements for 2NZ Div, and Aircrew for the RAF and Navy Personnel for the RN. It was about this time that the RNZAF and RNZN started operating in the South Pacific as well as the early elements of 3NZ Div.
I don't see the sense in shipping CKD vehicles from Canada or USA across the Pacific to be assembled in NZ, then shipping them across the Indian Ocean to Egypt. There were assembly plants in Egypt already.
Also the LP carriers do not have parts commonality with Universal Carriers.
We know that Aussie LP1 were taken to Greece and the Middle East, but also suffered from a lack of spares. Also Aussie LP2 and 2A were used in the South West Pacific (New Guinea, and New Britain).
The NZ LP1 were non armoured, so stayed home. And NZ LP2 were found ineffective in the Solomons, so the Carrier Platoons became infantry.
This covers a bit of ground but that's my reasoning!
Rob