According to a letter sent by Sara to Georgie, her great-grandson is the late Mr. Allen, Jamie and Lucy's father, so Jamie is considered heir to the house, being Sara Latimer's great-great-grandson.
In a letter from Sara to Georgie, it was stated that her great-great-grandson is the late Mr. Allen, who is also Jamie and Lucy's father. Consequently, Jamie is considered the heir of the house, being Sara Latimer's great-great-grandson.