Tommaso was trained in Naples and moved with his father and siblings (including singer Nicolina) via Graz (1747), Salzburg and Frankfurt (1750), Amsterdam (1752) and Paris (1753) to London, where they performed four burlettas at Covent Garden in the 1753–4 season.
He had a stake in the short-lived 'English Opera House', which he founded in 1783 and which produced works by Irish composers and librettists, also in a music shop, neither of which was financially successful.
Among his pupils were Lady Morgan, Thomas Simpson Cooke, and John Field, the inventor of the nocturne, who made his debut at one of Giordani's Rotunda concerts (4 April 1792).
Among Giordani's compositions are a number of operas and theatrical pieces, including Genius of Ireland (1784) and The Island of Saints, or The Institution of the Shamrock (1785), the oratorio Isaac (1767), and a large number of overtures, sonatas, concertos, quartets (mostly string quartets, though some with flute, keyboard or guitar), trios for violin, flute and basso continuo, songs, etc.
He is likely to have been composer and conductor at the Francis Street Chapel, Dublin, from approximately 1784 to 1798, and conducted a Te Deum of his own at the celebration upon the recovery of King George III, 30 April 1789.