Regnal numbers are very rarely used during the reign of the monarch they're referring to, as the general understanding was that if you said "The King" or even "King George", you were referring to the current king. This still holds true today; if a person says The Queen, with no prior context, you're likely to simply think of the current queen in your country of origin, if you have one.
However, regnal numbers are far from a new phenomenon. A book by Sir Francis Bacon, published in 1622, was titled "Historie of the raigne of King Henrie the Seventh". Another book, by Herbert of Cherbury, published in 1649, was entitled "The Life and Raigne of King Henry the Eighth".
It seems to largely be about distinguishing one monarch from another, as it is today. In 1650 a book was published, compiled by Sir Anthony Weldon, that was entitled, "The court and character of King James whereunto is now added The court of King Charles : continued unto the beginning of these unhappy times : with some observations upon him instead of a character". Due to the timing at which it was published, and the fact that, at that time, England had only had one King James, it seems it was unnecessary to specify. However, in 1656, Sir William Sanderson published, "A Compleat history of the lives and reigns of Mary Queen of Scotland, and of her son and successor James the Sixth, King of Scotland". In this case, because of the title referring to monarchs of Scotland, a regnal number was added to make it very clear which particular King James of Scotland it was referring to.
Ultimately, the answer is: George the third would probably not have been referred to as George the third during his reign, and instead would have been called King George, or simply, The King. However, immediately after his death, people may well have started referring to him as King George the Third (alongside The Late King George), to distinguish him from his son, George the Fourth, and his grandfather and great grandfather, George the Second, and George the First.