Uncle Nigel is Private's uncle.
Uncle Nigel is first mentioned in All Choked Up, where Private mentions while watching a documentary called "Carnage of the Penguins," that the penguin about to be eaten by a leopard seal looks just like his Uncle Nigel. However it was either not Nigel, or possibly a cover for disappearing, since he comes to visit in A Visit From Uncle Nigel.
In A Visit From Uncle Nigel, Nigel is shown to be an English spy looking for The Red Squirrel, the same one sought by Special Agent Buck Rockgut. His cover is that he loves to talk about sissy adventures and doilies as much as Private, and he congratulates Private on his own "cover". He leaves feeling very proud of Private - although Private's victories were due to luck and genuine sissiness, he is not taking credit for anybody else's victories, and simply agrees to Nigel's assumptions before Skipper says otherwise.