Melodrama began in France and Germany, arriving in England only after it had already been developed abroad. In France, a "melo-drame" referred to a work that used music and dumbshow, while in Germany a melodrama was a piece with spoken dialogue accompanied by musical underscoring.
It was in England, though, that melodrama developed, expanded, and was eventually exported all around the world. The Cambridge Companion to English Melodrama, edited by Carolyn Williams, aims to establish England's centrality in the emergence of the genre, containing essays contributed from top scholars, including Michael Gamer, Peter Brooks, and Juliet John.
Jim Davis has a great essay in the collection looking at the various theatres where melodrama was performed in England during the nineteenth century. Thomas Holcroft's A Tale of Mystery is often considered the first English melodrama, since Holcroft was the first person to use that term in describing his work, calling his play "A Mélo-Drame In Two Acts." It premiered at the Theatre Royal, Covent Garden in 1802.
Drury Lane that held royal patents. The West End theatre that really drew audiences in for melodramas was the Adelphi, previously known as the Sans Pareil when under the direction of Jane Scott. The theatre was demolished and rebuilt twice, once in 1858 and once in 1901. It was in the second building that Dion Boucicault staged his play The Colleen Bawn with it's sensational scene showing a woman being rescued from drowning.
Over in the East End, the Pavilion Theatre in Whitechapel opened in 1826 and served a working-class audience that typically included a number of sailors who walked over from the docks. Not surprisingly, the Pavilion was known for its nautical melodramas. It burned down in 1856, but was rebuilt two years later. During its earlier years, it provided a home for melodramas penned by Elizabeth Polack, including the biblically inspired play Esther the Royal Jewess.
On the south side of the Thames was the Royal Coburg, later renamed the Royal Victoria, and today known as the Old Vic. It had to compete with a slew of other theatres in the neighborhood, though, including the Surrey Theatre, which premiered Douglas Jerrold's play Black-Eyed Susan in 1829. The Surrey also produced Edward Fitzball's 1833 crime melodrama Jonathan Bradford, which featured a set that showed four rooms simultaneously.
There was plenty of English theatre outside of London, too, and Davis includes a section on provincial theatres that did melodrama as well. One, the Sheffield Theatre Royal, staged Joseph Fox's controversial melodrama The Union Wheel in 1870, causing an outcry in the press, with some critics considering the piece to be an incitement to strike.