I have not found what kind of business Ludstone & Son was involved in. However the street number -96- tells us this ghost sign was painted in the late 1930s or later. Indeed the section of Webber Street between Blackfriars Road and the Southwark Bridge Road and Great Suffolk Street junction used to be called Friar Street and street numbers only went as far as 58 on the south side (where the building is located). Between 1936 and 1939 several streets around London changed names. In the case of Friar Street, it was incorprated into an extended Webber Street and house numbers were changed accordingly. This would have been when this building was allocated the number 96.
Interestingly, while this part of London was already quite built up by the late 19th century, the small triangular plot of land between Rushworth Street and the railway viaduct was still unbuilt by 1915 if the edition of The Post Office London Directory published that year is to be believed.
Location: Webber Street / Picture taken in March 2009