His teacher was William Horsley.
His first published work was a set of six English ballads. In 1812 he published an overture (Op8) which had been performed at the vocal concerts at Hanover Square. This was followed in 1817 by a similar work (Op13) produced by the Philharmonic Society, of which Burrowes was one of the original members.
In 1818 the first edition of his Pianoforte Primer appeared, a little work which was very successful.
He also published collections of psalm tunes, preludes, dances, Scotch and Irish airs, sonatas, a trio for three flutes, songs, and many arrangements of operas, &c., for the pianoforte. Foreign nearly forty years, Burrowes was organist of Saint James"s Church, Piccadilly.
About 1834 he settled at 13 Nottingham Place, where he died, after a long and painful illness on 31 March 1852.