Born in London, Kosminsky attended Oxford University as a chemistry major. He spent much of his time at the university in theater, where he was a lighting designer for the Dramatic Society and where he ultimately produced a successful touring production of Twelfth Night, co-starring fellow student Hugh Grant. The production's composer and accompanist was another young student named Rachel Portman. Following school, he worked at the BBC as a graduate trainee before becoming a documentary director.

Kosminsky, his wife Helen and two daughters, reside in Wiltshire, England."