I don't think Thomas' brother William had a child with Amy. I think you are still looking for a William Hall, carter.
Adelaide could have lied at the time of her marriage and married as Broadhead and given Thomas as her father (many children in her situation did just that). In her case, she was very clear with everything and consistent! In the absence still of a birth reg for now, I would still go with what info she gave.
Need to keep looking for her birth reg and any clues for a possible father, William Hall.
Monica