There is no single legal definition of 'next of kin'. See here:
http://www.theanswerbank.co.uk/Law/Question441 871.html
As far as the situation would be if he was to die intestate, only blood relatives can inherit (in the absence, of course, of a spouse) but adoption counts as a blood relationship. So his wife's nephews and nieces would receive nothing. His own nephews and nieces (including by adoption) would receive a share of his estate but the shares might differ.
Let's assume (so that I can explain the mathematics) that the widower had one brother and one sister. If so, each of them theoretically gets half of the estate. As they're not alive to receive it, their shares are divided between their children. So if his sister only had one child, that child would get half of his estate. But if his brother had 5 children, each of his children would only receive one tenth of the estate.
Chris