As is noted a few lines later, the Old Mo was the Midddlesex Music Hall.
It was situated on what is now the site of the New London Theatre, on Drury Lane and Parker Street.
Known initially as the Mogul Saloon (an extension of the Mogul tavern), the Middlesex opened in 1851; its nickname derives from the original name.
Between 1910 and 1911, it was rebuilt under the name New Middlesex Theatre of Varieties; this building was demolished in 1965, and work began on the present theatre in 1971.
For a more detailed history, see here.