I think it would have been used when they made the organization post war in 1866 and used it last when the last soldier died in 1956. At the end of the war there was the grand review of the army but I've never read a letter or seen a period newspaper that referred to the GAR as the wartime name for the army. Union army and federal army and the theatre designations like army of the Potomac were used to my knowledge.
The first use seems to have been a June, 1861 speech in Washington, by Simon Cameron before the St. Andrew's Society, in which he concludes:
[Boston Evening Transcript, June 21]
After October, it is sometimes capitalized as "The Grand Army of the Republic." The use was widespread in the press (geographically), if not all that frequent. In one newspaper archive I find the phrase 59 times from 1861-1864 -- but almost always as a description of the army, not as its title.