I just noticed that my nursing villager is sick. Is there anything that can be done? I tried dragging the doctor on to her but nothing.

You should be able to heal a sick nursing mom the same way that you heal any other sick villager. Try again with your doctor, and if that doesn't work, try healing her with a different villager (even if that villager has no skills as a doctor). Sometimes villagers resist the healing efforts of one villager, but respond to a different villager.