Formerly a house, now a nurses' home, according to Historic England. Seems to add up, as it appears to now be on the grounds of the Bristol Homeopathic Hospital, which is "being used as the student health service for Bristol University"
On the left of the door is the service wing, apparently. It's a sprawling old place, and seems very awkwardly shoehorned into this corner. It looks from old maps like it was backing onto a quarry for a while, before that was filled in and the homeopathic hospital was built there.
GPS Coordinates:
51.46241, -2.60477
Location: Tyndall's Park