The present-day Her Majesty's Theatre was opened in 1896 (for part of its history, it was known as His Majesty's Theatre).
It is the fourth theatre to have existed on the site, previous incarnations being known (at different times in their history) as The Queen's Theatre, The Italian Opera House and The King's Theatre.
The third theatre to be built on the site was also known as Her Majesty's Theatre; it was opened in 1869 and demolished in 1892. This is the theatre which Nan and Kitty would have seen on arrival in London.
Follow this link for more information, and to see pictures of the various theatres which have existed on the site.