Joshua was born in Cardiff and studied at the Royal College of Music. After completing her studies there, she made her debut at the Aix-en-Provence Festival as Angelica in Handel's Orlando. Other early operatic appearances include Zerlina in Don Giovanni with Scottish Opera, Susanna in Le nozze di Figaro and Sophie in Der Rosenkavalier for English National Opera. Following this, she has sung a number of major soprano roles in many of the world's leading opera houses and festivals, including Adèle in Die Fledermaus at the Metropolitan Opera, New York; the title role in The Cunning Little Vixen at La Scala, Milan and De Nederlandse Opera in Amsterdam; Anne Trulove in The Rake's Progress at the Glyndebourne Festival, and Zerlina at the Royal Opera House, Covent Garden.
She has appeared in concert across Europe and throughout the UK. Her discography includes the title roles in Handel's Partenope and Semele, Sophie in highlights from Der Rosenkavalier, and the solo in Mahler's 4th Symphony with Philippe Herreweghe conducting the Orchestre des Champs-Elysées. In 2012 she recorded Harmonia Sacra, a collection of lesser-known spiritual songs composed by Henry Purcell.