Crosdill's musical training began when, at a young age, he became a chorister at Westminster Abbey under the direction of John Robinson and Benjamin Cooke.
Crosdill made his first public appearance as a violoncellist at age nine when Emanuel Siprutini (1730–1790), his probable teacher, presented him in a duo.
In 1782, Crosdill was appointed chamber musician to Queen Charlotte and violoncello instructor to the Prince of Wales (later George IV).
Crosdill played in the annual productions of Messiah and was principal violoncellist for the 1784 Handel commemorative concerts given at Westminster Abbey.
Elizabeth was many years Crosdill's senior and her comfortable income from interests in Wiltshire, allowed him to retire from public performance.
By his will signed at 45 Berners Street on 30 August 1825 and proved at London on 25 October 1825, his son from a subsequent marriage, Lieutenant Colonel John Crosdill of the East India Company, inherited a substantial fortune.
Crosdill left bequests of 19 guineas each for remembrance rings to numerous friends including the musicians Benjamin Blake and William Shield, his Berners Street neighbour.