Gentleman's Magazine 1751
This is not official. One needs to find the London Gazette, from which this is taken, but it says, after calling him by his titles which were his by birth or inherited from his father (i.e., Prince of Great Britain, Electoral Prince of Brunswick-Lunenburgh [sic], Duke of Edinburgh, Marquis of the Isle of Ely, Earl of Eltham, Viscount of Launceston, and Baron of Snaudon [Snowdon]), that he was created Prince of Wales and Earl of Chester. Thus not created Duke of Cornwall or of Rothesay, etc.
Cokayne, in The Complete Peerage, agrees that George III was not Duke of Cornwall, saying that since he was not "filius regis" he was not entitled to that title. He goes on to say that George III is the only instance of an heir apparent not possessing the title "Duke of Cornwall". There was one previous instance of a grandson being heir apparent, that of Richard II, but Richard II had been CREATED Duke of Cornwall, and did not obtain it by right, as all other Dukes of Cornwall did.