The double chimney pot on the roof (marked by the red cross) could not be related to a fireplace in the children's bedroom.
The White side chimney of the double chimney may not connect to a fireplace in the children's bedroom now, but that doesn't mean it didn't when originally constructed. The kitchen Aga has a quite separately located chimney that is also visible in the photograph.
By the way, how was the house heated? I haven't seen in the photographs any fireplace that seemed to be in use, or any radiator or other heater (except the Aga).