Browne was born in Dublin, sentenced to transportation in 1810 and arrived in Sydney in 1811 on the Providence. Within a few months of arriving, he reoffended and was removed to the secondary penal colony of Newcastle. In Newcastle, Browne came into contact with Lieutenant Thomas Skottowe, the commandant of Newcastle from 1811-1814. Skottowe was interested in natural history and commissioned Browne to create drawings of his collections to illustrate a manuscript entitled, Select Specimens From Nature of the / Birds Animals &c &c of New South Wales, Collected and Arranged by Thomas Skottowe Esqr. The Drawings By T.R. Browne. N.S.W. Newcastle New South Wales 1813. Browne’s illustrations of insects are particularly fine.
Until 1817, many of Browne’s works are signed with an extra initial, usually J, I or T. Like Joseph Lycett, Browne contributed many of the original watercolours for Major James Wallis' An historical account of the Colony of New South Wales which were engraved by Philip Slaeger (Sligo) and Walter Preston.
After 1817, when he gained his freedom, Browne returned to Sydney. His illustrations from this period concentrate on the Indigenous peoples of the Sydney area. Several of these are included in Collection of portraits, predominantly of Aborigines of New South Wales and Tasmania, ca. 1817-1849. He died in Sydney.