On the opening night of Princess Ida at the Savoy Theatre in January 1884, the renowned composer Sir Arthur Sullivan (Jim Broadbent), who is suffering from kidney disease, struggles to reach the theatre in time to conduct. Hoping that taking a holiday on the continent will benefit his health, Sullivan takes some time off. However, during his absence, the sales of tickets and attendance at the Savoy Theatre dwindle under the oppressive summer heat. Producer Richard D'Oyly Carte finds himself in a difficult position, having called upon both Sullivan and dramatist W. S. Gilbert to create a new production for the Savoy, but unfortunately, their new work is not ready when Ida concludes. As a temporary solution, Carte decides to revive an earlier work by Gilbert and Sullivan, The Sorcerer.
As Gilbert begins crafting ideas for their next opera, he proposes a magical lozenge that transforms its users. Yet, Sullivan feels this idea is overly reminiscent of the magic potion and various other supernatural elements utilized in their previous operas, making it feel mechanical. Constrained by the expectations of the British music establishment to produce more serious compositions, Sullivan yearns for something "probable," focusing on "human interest" and eschewing the reliance on magic. Gilbert, however, believes his libretto is sound and stubbornly refuses to create a new one, leading to a creative impasse.
The friction is finally resolved after Gilbert and his wife attend a popular exhibition of Japanese arts and crafts in Knightsbridge, London. A moment of inspiration strikes when a katana sword he acquires unexpectedly tumbles from the wall of his study. This incident sparks his imagination, leading him to craft a libretto set in exotic Japan. Sullivan embraces this fresh concept and eagerly agrees to compose the accompanying music.
The trio of Gilbert, Sullivan, and Carte immerse themselves in the preparations to ensure The Mikado becomes a triumph. A flurry of activity ensues as scenes of rehearsals and behind-the-scenes stresses unfold: cast members enjoy lunch together while attempting to negotiate their salaries. Gilbert seeks assistance from Japanese girls at the exhibition to coach the ladies' chorus in authentic movements and fan techniques. The main cast undergoes costume fittings, created by the talented C. Wilhelm. Tension rises when the cast disputes Gilbert's intended cut of the titular character's Act Two solo, “A more humane Mikado,” ultimately persuading him to restore it.
As the long-awaited premiere approaches, the actors experience first-night nerves in their dressing rooms. Finally, after much anticipation, The Mikado is set to open. True to form, Gilbert's nerves get the best of him, causing him to pace outside instead of watching the performance. After a nerve-wracking wait, he returns to the theatre to discover that the new opera is a resounding success.
