Hi again Mare
Well I guess we should be none too surprised at learning that the "issue" column on Charlotte's death cert., shows as a blank ? More positively though, further "confirmation" that Charlotte and Aner COOK have arrived in NZ around 1861.
It appears that the child Aner COOK - born London - 1861 to Abraham Robert and Anna Maria COOK, has died.
FREE BMD UK shows a birth registered for "Aner COOK - June quarter - 1861 - St. Lukes"; and a death in the same registration period, also St. Lukes.
[ Aner COOK's baptism, (on IGI), just 12 days after his birth,
may hint that this child was not expected to live long] ?
If perchance, there remains 20 bucks in the kitty, Mare - (yes, an expensive "hobby" this is) - the death "printout" for Walter Robert COOK (1923), would be worthwhile obtaining ? To confirm - (a) his parentage, (we're only speculating at present, that the chap in Brighton / Charleston, is the nephew of Aner and Charlotte) .... and (b) "Years in NZ". If he has arrived after say 1870, "he's off the hook", BUT, if prior to that, then there's the possibility he may have fathered an "illegitimate" child? (Perhaps a child "adopted" by Aner and Charlotte) ?
Who know's what his death cert. might reveal

Some kindly, fellow RootsChatter, may also do for you, a look-up of the probate record for Walter Robert COOK - 1923, held at Archives, Christchurch ?
Lu