Unless you've been able to access some particularly specific records, it is unlikely this information is easily accessible on the internet. As someone who has a strange obsession with the history of girls private and boarding schools (as a result of picking up a book Ruby recommended, funnily enough) records are not digitised very well. For instance, for one school that no longer exists I had to scrape local newspapers to find the list of headmistresses to add to the Wikipedia article for the school once. No one had thought to put it together in an easily readable source before I decided to, and I see this quite often with non-digital resources on obscure schools.
There were countless small rural schools for girls, situated much like Bones manor. These school buildings were not purpose built as such, rather the headmistresses purchased them. Many people gave up their large houses during the 20s, 30s, and 40s, as they could not afford to run them (staff became increasingly expensive over this time.) Often, and annoyingly, these schools would simply go by the name of the house, and would change name if they changed premises (many a school "disappears" due to this in the records). Many of these schools closed well before the internet came along, and hence no one has bothered to record them existing online. Reasons for closing are myriad: some ran out of pupils, others just out of money, some out of staff. Many schools closed when their headmistresses died, with no one willing to step up and take on the running of the business.
It is possible the house, then, held both lives: as a small boarding school and as a country pile. I think it unlikely that the water fountain has anything to do with a previous life as a private residence, rather would come from the time as a school if it had one. I've never heard of an employer bothering with a water fountain for their staff (or anything similar).