In a quite lovely bit of sleuthing, the Oxford English Dictionary has discovered a use of"OMG" from 1917 [1]. It comes in a letter by British Admiral John"Jacky" Fisher, who wrote that:

I hear that a new order of Knighthood is on the tapis—O.M.G. (Oh! My God!)—Shower it on the Admiralty!

The letter was published in his Memories, published in 1919.

Fisher was famous for being the driving force behind the creation of the HMS Dreadnought, an advanced capital ship which, when it was launched in 1906, seemed revolutionary. This, the world navies agreed, made all other capital ships obsolete, but, distressingly to the British, destroyed their long-standing lead in naval power, if temporarily. The result was an enormously expensive Anglo-German naval race, which did much to bring on World War I.

Fisher was also famous for his wild flights, often expressed in intemperate letters. In 1915, as First Sea Lord of the Admiralty, he and Winston Churchill (then head of the Admiralty) clashed, and Fisher, on May 15, 1915, resigned his position in a note to Churchill,"I am unable to remain any longer your colleague...I am off to Scotland at once so as to avoid all questionings."

Churchill, somewhat flummoxed by the idea that the uniformed head of the Navy was simply abandoning his post, found the Prime Minister H.H. Asquith and related the story. Hoping that Fisher had not yet left, Asquith wrote a terse note to Fisher--"Lord Fisher, In the King's name, I order you at once to return to your post, H.H. Asquith, 15 May 1915"--and sent his aides to track down the Admiral. The Prime Minister might, one thinks, have been forgiven for using"OMG!" himself at that moment, but Asquith was always more restrained.

[1](sub. required)"OMG, int. (and n.) and adj.". OED Online. March 2011. Oxford University Press. http://www.oed.com/viewdictionaryentry/Entry/293068 (accessed March 24, 2011).