Mare,
This is very exciting. Our branch of the family has been investigating the Cook whanau for a while, spurred by those "Who Do You Think You Are?" programmes.
A random Google search brought up your thread about William Robert Cooke. I gather that William's parents were Aner and Charlotte Cook (nee Smart). I had been investigating those very people, but apart from a reference to them in the 1861 England census they appeared to have been lost from English records. I concluded that they had in fact emigrated, and was delighted to find that this was the case. Thank you for posting!!!
In the 1861 England census, Aner is actually "Amos" in the records, born 1814 in Wilby, Suffolk and living in St Lukes, Middlesex with his wife Charlotte. At that point they have no children (apparently).
As they do not appear in subsequent English census records, "Brighton" on William Robert's birth certificate is likely to in fact be Brighton, New Zealand. I think it strongly likely, as Walter Robert Cook was born in Brighton, New Zealand in 1876.
Returning to the 1861 census records - Abraham Robert Cook and wife Anna Maria (nee Smart - the sister of Charlotte) are Aner and Charlotte's neighbours. I suspect Abraham and Aner are brothers but have not been able to prove the link yet. Charlotte and Anna are certainly sisters - their parents are William Smart and Frances (nee Gibbons/Gibbins). They also had two brothers - George and Henry.
Abraham and Anna had seven children altogether, all of whom, apart from the first, were born in London. I can give you their names if you like... What you'll probably be interested to know is that Walter Robert is one of them - born 1847, St Lukes, London. Walter Robert (or WR1 as we call him) came out to New Zealand in 1873 (we estimate). He married Elizabeth (maiden name an issue - we have 5 different spellings) and had nine children (Maria Murphy, Frances Ellen etc, etc).
So.... Walter Robert and William Robert were in fact cousins (if Abraham and Aner are brothers which seems likely). And it explains why Walter Robert settled his family on the West coast - makes sense if Aner and family are already here.
I can tell you as much as you want to know about Walter Robert (WR1) - he is my great, great grandfather.
As I said - delighted to find your post!!! I think that makes us cousins umpteen times removed or something... If you have any more info, or would like more info from me then get in touch.