Wilde was born in New York City, New York. Her mother, Leslie (née Corkhill Redlich), is a 60 Minutes producer and journalist, and her father, Andrew Cockburn, is an Irish journalist, as are her uncles Alexander and Patrick, all of whom are contributors to the political website CounterPunch.org. Her aunt is mystery writer Sarah Caudwell. Her paternal grandfather was the British novelist and journalist Claud Cockburn. Wilde has said that as a result of her family background, she has a "strong journalistic streak", being "really critical and analytical".

She has wanted to become an actress since the age of two. For a short time, Olivia's family had a house in Guilford, Vermont, USA. Wilde attended Georgetown Day School in Washington, D.C., as well as Phillips Academy, and subsequently moved to Dublin, Ireland, where she attended The Gaiety School of Acting.[1] She initially began working as a casting assistant.