One of my rellies was in Fairmile in early 1900's she was buried in Cholsey churchyard. Might be worth a try. If her family was well to do, they mght have arranged to have her buried in their local parish.
Sorry, I have the wrong rellie, have rechecked, the one who was definitely a patient, it says County lunatic asylum by the entry for church. The one who was a bit "touched" did not die in the asylum, it says St Mary's by entry for church. Her death was somewhat later and one of the early graves in the new section of the churchyard. I saw it some years ago.