Geoghegan began studying the bassoon at the age of twelve, having previously played the violin since the age of five. She later studied at the Junior Academy of the Royal Scottish Academy of Music and Drama. She won the Gilbert Innes Prize for woodwind and the 2004 Concerto competition, and in 2004 and 2005 was awarded the Wolfson Scholarship. In 2006 she gave a recital on BBC Radio Scotland, and also performed the Hindemith Trumpet and Bassoon Concerto. She was a member of the National Youth Orchestra of Scotland for four years and in 2007 performed with Camerata Scotland and NYOS Futures.
As of 2009, Geoghegan is in her final year at the Royal Academy of Music with John Orford. During her first year she performed the Mozart Bassoon Concerto with the Academy String Orchestra, and received the Violet M. Wallace Award. She was runner-up in the BBC Two programme Classical Star for her performance of the Hummel Bassoon Concerto with the City of London Sinfonia at LSO St Luke's. On 5 August 2009 she performed the Mozart Bassoon Concerto with the BBC Philharmonic at the BBC Proms.
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