There were girls boarding schools in regency times - I've recently been re-reading Jane Auten's 'Emma' and Emma's friend, Harriet Smith is at a boarding school in the village, with the owner of the school being one of the ladies who can be relied upon when Mr Woodhouse wants people to come to play cards with him of an evening.
In her description of the school, which I think is described as having about 40 pupils, Austen describes it as an 'honest, old fashioned boarding school' and makes a few digs at other schools, so I guess they were pretty common.
I would have thought that it was more likely to be middle class girls who would go, daughters of sucessful merchants etc rather than those of the landed gentry (going back to Austen again, there's no suggestion of any of her heroines going away to school, he assumption seems to be that you either have a governess (with additional tutors for things such as music, dancing or drawing as required) or are taught by your own mother. (remember Lady Catherine de Bourg's comments about Mrs Bennett must have been a slave to her daughters' education, when told they had no governess?)
An illegitimate child might well be placed at school, by her father, to give her a better chance than if she stayed with her mother.
I would have expected that the orphaned child of an aristocrat would be more likely to have an establishment formed for her, either in her own house, or elsewhere, rather than be sent to school. Her guardian or nearest relation might well either take her to live with them, or move into her own property to care for her.