"Union Street is opposite Pipe Lane, and leads from North Street to Donegall Street. It was said to have got its name from having been first opened at the time of the Union, which can scarcely be correct, as it is called Union Street in the map of 1792" (George Benn, A History of the Town of Belfast, vol. i, 1877, p. 530).
The IHTA states that a 1790 map of Belfast by Hugh Smyth shows Union Street. As the word 'union' can be used in so many contexts, the exact origin of this name is very problematic and it has remained something of a conundrum since Benn's time. Perhaps it is connected with the Belfast Union Volunteers, formed on 12th June 1778, apparently so named because they came about through the merger of two pre-existing companies of Volunteers. Their uniform was scarlet, faced blue, and they were commanded by Captain Lyons (F. J. Bigger, 1909, 'The National Volunteers of Ireland, 1782' (cont.), UAS ser. 2, vol. 15, no. 4, pp. 141-48. This is available on JSTOR).
According to Patton, the portion of Union Street from North Street to Little Donegall Street was opened in 1788.