James the baby is played by 4 different, unrelated babies. So if the hair changes, even from scene to scene, it's because child labor laws limits the amount of time a child can be in front of the camera and under the lights. It's entertainment. Suspend your disbelief.
Doctor Ellingham tells a patient that he'll need "regular doses of ascorbic acid, citrus and vitamin C tablets." In fact, ascorbic acid is vitamin C, so why mention it twice?
During the first part of this story, Martin's and Louisa's baby had a downy baby head of hair, but towards the end he had changed to a full head of blond hair.