Finch was born in Llanon, Ceredigion, and began learning the harp at the age of six. Her mother is German and her father English, but she is a fluent Welsh speaker.[2] By the age of nine, she had passed her Grade VIII harp examination.[3] She was a member of the National Youth Orchestra of Great Britain at the age of ten, becoming the youngest of its members to play at The Proms.[4] She studied harp with Elinor Bennett, who would become her mother-in-law, before attending the Purcell School, a specialist music school for children in Hertfordshire. She continued her studies at the Royal Academy of Music in London where she studied harp with Skaila Kanga.[5][3][6]
During the 1990s, Finch won several competitions for young harpists, including the Nansi Richards Prize and the Blue Riband at the National Eisteddfod of Wales.[5]
In 2003, Finch married the music and television producer Hywel Wigley, son of the former Plaid Cymru leader Lord Dafydd Wigley and the harpist Elinor Bennett.[6][15] They have two children together. The couple formally separated in 2017.
In February 2018 Finch announced on that she had been diagnosed with Grade 3 breast cancer.[16] She was given the all-clear in October 2018 following treatment at Velindre Cancer Centre, Cardiff.