The OED gives this as British slang of unknown origin for an old-fashioned and ineffectual old person and cites 1904. Other sources, because of earlier citations in American newspapers from 1889 onwards, maintain that the expression is originally from America but possibly via Scottish origins. In the 19th century, the expression ‘duddy fuddiel’ was used in Scotland to describe a raggedy, dishevelled person. There might be a link from this to fuddy-duddy but it cannot be proved. That is why the OED’s verdict is origin unknown.