I checked Tamiya's catalog, because they have produced so many WWII kits in 1/35, but they don't have a British staff car, either.
As far as anyone producing one goes, I wouldn't hold my breath. Only if a company thinks it would sell, would they add one. There probably hasn't been much demand for a British staff car.
A web search on "wwii british staff car" returned a good number of hits, including a photo of Montgomery's Humber. A lot of the photos look like the British Army used a lot of civilian cars, repurposed, like the US Army did. Have you tried looking for models of the cars in civilian livery, that you could repurpose for military use?