It seems that the Sunbeam 16 and 24, (1915-1924) is the subject of a recently published book, (ISBN: 978-0957420809). This is reviewed in the Automobile, (June 2013, volume 31, number 4), which refers to 4000 Sunbeams built by the Rover Co. Ltd. under a wartime agreement.
With so little information yet published on staff cars of that era, it does at least seem that material may exist.
An attractive design, a model of a British WW1 motor ambulance wouldn't go amiss, as far as i'm aware there's only the Matador Ford Model T which needs work, and these are quite a lot larger
Well, those tyres, which may be larger than originally intended, andof a later hue. It does seem that light coloured tyres are sometimes available, as I have seen Model T Fords so equipped.