Sangster was born in London. He has a sister, Ava, and is the son of actors Anastasia "Tasha" (Bertram) and Mark Ernest Sangster.[2][3] His father is a musician and film editor.[3] Sangster is the second cousin, once removed, of actor Hugh Grant, alongside whom he appeared in Love Actually. His maternal great-grandmother, Barbara (Randolph) Bertram,[4] and Grant's grandmother were sisters.[3] Sangster's great-grandfather, Anthony Bertram, was a novelist, and one of Sangster's maternal ancestors was politician and colonial administrator Sir Evan Nepean.[5]
Sangster plays bass and guitar, and he learnt to play left-handed guitar to portray the left-handed Paul McCartney in the feature film Nowhere Boy.