Hi Sue
I've re-checked NZ births for years 1910 - 1911- 1912 using the "01 January" birthdate you gave and there isn't anything that matches. Of the results returned, I was able to eliminate them all.
[My note: Because a birth registration/or record, has not as yet been located, bear in mind that the "1st of January" (in any year), is very often used as a "default date" (- where actual date is unknown). ] -------
...... I have just messaged my cousin NZ and ask for John Birthday .....
..... We thought she had found a John Yeo that could be Mr Anderson but I found a tree on familysearch sadly not him. 
.....
..... I think she has been looking a long time and keeps getting dead ends. .....
I just ask the following because it seems that the info you have to date re: John ANDERSON, comes mainly from "indexes" or "published family trees" (as opposed to official records) ??
Questions:
* Are you doing this "John ANDERSON" search in conjunction with your "NZ cousin"
OR ... doing separate searches and then comparing "notes"
* Does "NZ cousin" have any
"actual official records"(WW2 Army record / marriage/ death) for John ANDERSON ?
[Incidentally, her own full NZ birth record, will list the birthplace of her father and his age at that time. Useful for comparison with other records. 
] -- e.g. His 1955 NZ marriage record ?
[This record -
in printout* format (* which carries more information than a certificate) - would be the best starting point in reinvigorating a search which has reached a
dead end. John ANDERSON himself would have supplied details, amongst which (if known) were > "the names of both his parents / his father's occupation / mother's maiden name ... his own place of birth." ]
Again, as a record to which John ANDERSON has himself contributed details, his WW2 NZ Army file would be a very useful resource.
And because of the time spans which separate the creation of these particular records > , Army enlistment, circa 1939 ... marriage some 15 years later ... there may be quite contrary information (as there will possibly be too, with details given by a third party for his 1986 death record). But somewhere within, there might be a vital "clue" that allows you to move onwards.
Probably not answered sufficiently at outset of this thread, but New Zealand has a great array of useful records available > all very well documented in the "Resources section of this New Zealand board". [ NZ BDM records in particular contain far more info that those of UK. ]
https://www.rootschat.com/forum/new-zealand-resources-offers/But you'll understand that not every record can be had for "free" via the internet and, in the case of John ANDERSON, his "dates" dictate what is considered both "historical and non-historical data" - and the processes for obtaining (NZ) records of same, differs. [e.g. His 1955 marriage record ($) is classified as "non-historical" -- his WW2 file (not yet online), but available (at no cost?) by application form at the NZ Defence Force website -- Death record readily available ($) from NZ BDM site. ]
~ Lu
Edited for clarity ( x 2 ) 