Southwark and Deptford Tramways
(from 1893, London, Deptford and Greenwich Tramways)
Summary
Although only a small number of photographs of this south-east London horse-drawn tramway (1880 to 1904) have survived, these are enough to state with a reasonable degree of certainty that drivers wore robust but informal attire, along with the fashionable headgear of the period, the bowler hat.
The only surviving photograph of a conductor taken prior to 1893, shows the subject in a long single-breasted overcoat, possibly self-purchased; it is unclear whether these coats carried any insignia, though probably not. Caps were in a kepi style and do not appear to have carried a badge, at least one that shows up photographically. Late in the tramway's life, conductors certainly wore double-breasted uniform jackets with two rows of composite (i.e., non-metallic) buttons and lapels, and military-style caps with a tensioned crown (top). Neither the jackets nor the caps carried insignia.
Drivers and conductors always appeared in service with an enamel Public Carriage Office licence (see link), which was usually hung from a jacket button, or, in the case of conductors, from the cash-bag strap.
There is one newspaper reference to a court case involving a London, Deptford and Greenwich Tramway Company inspector, so the company certainly employed more senior grades, however, as no photographic evidence has survived, it is currently unclear whether or not they were issued with uniforms.
For a history of the company, see 'London County Council Tramways Vol 1: South London' by E R Oakley; London Tramways History Group (1989).
Images
Horse tram drivers and conductors
Tramcar No 25 of the Southwark and Deptford Tramways Company stands in the depot yard on Evelyn St — photo undated, but probably taken in the late 1880s or early 1890s. Photo courtesy of the Tramways and Light Railway Society, with thanks to David Voice.
A blow-up of the above photo showing the conductor, who appears to be wearing an overcoat and a kepi-style cap.
Another blow-up of the above photo, this time showing the driver, who is wearing smart but informal attire. The oval shape on his coat is almost certainly a Public Carriage Office licence. 
Unfortunately out of focus, but nevertheless of interest given its rarity, is this relatively late shot of a London, Deptford and Greenwich Tramway Company horsecar and crew, probably taken around the turn of the century. Photo courtesy of the Tramways and Light Railway Society, with thanks to David Voice.
A blow-up of the above photo showing the driver (on the platform in informal attire and bowler hat,) and the conductor (at the front), in double-breasted uniform with plain composite buttons and a military-style cap, seemingly, though not conclusively, without a cap badge. There is a possibility that this man could in fact be an inspector, though the style of uniform would suggest not.