There seems to be some confusion here between 'next of kin' (as a general term) and the specific rights which apply when someone dies without leaving a will. I'll try to provide some clarification:
If a person dies without leaving a will there are strict laws which clearly define who will inherit their estate. They're shown in this flowchart:
https://www.wrighthassall.co.uk/media/article/attachments/Intestacy_flowchart_after_1.10.14_5_Aug_14.pdf
There are also rules defining a 'pecking order' of who is entitled to seek 'letters of administration', entitling them to distribute their deceased person's estate,. The order is as follows:
the deceased’s spouse or civil partner ;
the deceased’s children ;
the deceased’s parents ;
the deceased’s brothers and sisters 'of the full blood'.
So, in the circumstances described in your question, it's ANY child of your cousin (or two or more of his children jointly) who can seek a 'grant of representation' (in the form of 'letters of administration') if he died intestate. (It's NOT just the eldest child). However if he left a will then it's the EXECUTORS of that will who must seek the 'grant of representation' (in the form of 'probate')
However 'next of kin' has little, or no, meaning elsewhere. If someone goes into hospital for an operation they can nominate whomever they like as 'next of kin' on the forms. When someone dies there is no 'next of kin' rule about who should organise their funeral. (Indeed, there's absolutely no obligation upon anyone to organise the funeral. If everyone refuses, the local authority must do so).
At times it might be necessary for doctors (or carers) to work out who has the right to determine the type of care given to a person, thus forcing them into deciding who counts as 'next of kin' for such purpose. However there are no strict rules and, unless a court is asked to provide a ruling, it's entirely down to 'common sense'.