Disney was educated at home until the age of 16, when he went up to Peterhouse College, Cambridge. In 1798 he was admitted to the Inner Temple, and was called to the Bar in 1803. Subsequently he was appointed Recorder of Bridport in 1807 and Sheriff of Dorset in 1818. He left the post of Recorder in 1823 and at some point afterwards moved to Essex.
In 1816 his father died and Disney inherited a collection of antiquities that had been collected by his family. He began to catalogue these antiquities, and to add to the collection. The collection was further enlarged by gifts from Charles Callis Western and James Christie. His catalogue was eventually published in 1846 as Museum Disneianum, closely followed by a second edition and supplements.
Disney presented most of the sculptures in his collection to the Fitzwilliam Museum, Cambridge in 1850. In 1851 he endowed with £1,000 the Disney Professorship of Archaeology at the University of Cambridge. Upon his death in 1857 an additional £3,500 were bequested to support the Disney Professorship, and what remained of his private collection was sold by Christie's auction house.
Disney was married to Sophia Disney-Ffytche in 1802. They had three children; two sons and a boy. Disney was elected Fellow of the Royal Society in 1832 and Fellow of the Society of Antiquaries in 1839.