He was educated at King"s College in Windsor, went on to study law and was called to the bar in 1810.
He represented Cape Breton County in the Nova Scotia House of Assembly from 1820 to 1830. Uniacke went to Cape Breton Island, then a separate colony, in 1813. He was named to the Executive Council and became acting attorney general.
From 1815 to 1816, he served as the island"s acting chief justice.
He resigned his appointments in Cape Breton and returned to Halifax where he became advocate general of the vice admiralty court in 1819, succeeding his father. On July 21, 1819, he took part in the last known fatal duel in Nova Scotia.
William Bowie (merchant), a Halifax merchant, was fatally wounded (and later buried in Old Burying Ground). Uniacke and his second, Edward McSweeny, were charged with murder. they were prosecuted by Samuel George William Archibald but were acquitted.
In 1820, he was elected to the legislative assembly for Cape Breton, after it was reunited with Nova Scotia.
Uniacke was named King"s Counsel in 1824. In 1830, he was appointed a puisne judge of the Supreme Court of Nova Scotia. He died in Halifax in 1834, reportedly having become deeply depressed after sentencing two men and a woman to death the year before for the murder of the woman"s husband.