Final answer:
The narrator believes her room was initially a nursery and later used as a playroom and gymnasium.
Step-by-step explanation:
The narrator believes her room was once a nursery. This is derived from the text describing the room as having been "a nursery first" followed by its use as a "playroom and gymnasium", as suggested by the barred windows for the safety of little children and the presence of rings and other fixtures for physical activities.