Bulwer was born at Aylsham in Norfolk and studied at Jesus College, Cambridge. During his time at Cambridge he took drawing lessons from the famous landscape artist John Sell Cotman, and became a fellow of the Linnean Society due to his interest in mollusks, one of his three proposers being William Elford Leach.
In 1818 he was made a deacon and in 1822 a priest. In 1823 he became curate of Booterstown in Dublin, moving to Bristol in 1831 and St James's Chapel, Piccadilly in 1833. He spent several winters travelling in Spain, Portugal and the Madeira Islands, sometimes in the company of the philosopher and traveller Alfred Lyall. In the spring of 1825 Bulwer collected a specimen of an unknown petrel in the Madeira Islands. This was described by William Jardine and P. J. Selby in 1828 and given the common name of Bulwer's Petrel. The petrel genus of Bulweria was also named for him.
Bulwer left London in 1839 and moved back to Norfolk, becoming curate of Blickling and later Hunworth. He renewed his acquaintance with Cotman when his sons attended King's College School, and several of his sketches of Spain and Madeira inspired Cotman's watercolours.
- Barbara and Richard Mearns - Biographies for Birdwatchers (1988) ISBN 0-12-487422-3