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.
According to a letter from Sara to Georgie, her great-great-grandson is the late Mr. Allen, who is also Jamie and Lucy's father. Therefore, Jamie is deemed the heir to the house, as he is Sara Latimer's great-great-grandson.