An American slang term used for referring to oneself. "If you have any problems, just call your Uncle Dudley" is a way of saying "if you have any problems, just call me". Now rather out of date, although still occasionally used.
This comes from a style of editorial writing that was used by the Boston Globe, starting in the 1890s. These editorials were essay-like, somewhat informal, affable, and philosophical. They were always signed Uncle Dudley, no matter who wrote them. "Take it from your Uncle Dudley" became a popular saying.
There was also a movie released in 1935 entitled Your Uncle Dudley. All I know about it is that it was a comedy starring Alan Dinehart, Edward Everett Horton, Lois Wilson, Rosina Lawrence, and John McGuire.