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Wow - I know this was a couple of years ago but I have just 'found' a photo taken by Stuart Bros. I'm not sure who the pic is of but at least the company name has narrowed the time frame so thanks.

PeteFeet I can't believe that William Stuart originally came from Sidbury.  One branch of my family also came from Sidbury - Saunders/Raymonds.  I photo I have I believe is of a possible sister-in-law to the Saunders branch.  It's such a small world - especially around Sidbury   

Hope you're still interested in the no 3 hospital in Exeter.  I've just registered with this site so I can tell you what I know about this hospital. 
The building was attached to the workhouse and was a childrens home before WW1.  The military took it over during the war.  There are photos of ambulances and the outside of the building on local Exeter sites.  I would recommend that you look at Exeter Memories as it gives the hospital/workhouse history.  The building is still in use by the local NHS Trust as their HQ for Occupational Health. 
The home/hospital is part of the Heavitree Hospital site.  Most of the workhouse was destroyed during WW2 on May 4/5 1942 but the childrens home appears to be undamaged.
On a slightly different thread to your query, my Grandmother lived in a pub near the railway station and told my mother how she would watch the injured soldiers walk up from the station.  They walked in a line, one hand on the shoulder of the man in front.  There seems to have been a large number of military hosps in Exeter.
As far as the records are concerned I'm not sure where they would be but perhaps you should look up the online catalogue for the Devon Records Office.  I will say that they are very helpful.
Hope this is of interest to you
Regards
Alison