He emigrated to Australia in 1949, studied at the NSW State Conservatorium of Music, and gained a Diploma in piano in 1960.
Cuckson followed this with private studies in piano, composition and theory, in the UK and the US, his teachers including Ilona Kabos and Carlo Zecchi (piano), Georg Tintner and Peter Racine Fricker (composition), and Allen Forte (theory).
Cuckson has held various positions at the Mannes College The New School for Music in New York City since 1971.
Since 1991 he has also taught in the Theory of Music Faculty and the Keyboard Studies Program at the Curtis Institute in Philadelphia.
Cuckson has completed two operas, The Blind Men and Adrian and Jusemina, based on plays by Michel de Ghelderode (translated and adapted by the composer), and has written numerous other works for a variety of media: orchestral pieces, chamber works, piano solos and songs.