The Guardian doesn’t have the nickname “The Grauniad” for nothing, a paper likely to spell its own masthead wrong.
The nickname Grauniad was reportedly coined by the British satirical magazine Private Eye. This seems to be confirmed by the following, from The Economist (London) of 27th November 1971:
All the characters of Private Eye’s earliest days survive—Lord Gnome, the Grauniad, Lunchtime O’Booze—giving a curious cosy familiarity to each issue.