As far as scientific research goes saffron's relation to skin complexion is a myth. Remember your baby cannot have blue eyes and blond hair or look like Chinese if neither of the parent carried that gene - same is true with skin - you can only pass on what you have. But there are other medicinal properties of saffron.
Surprising Saffron - Health.com http://www.healthrecipes.com/saffron.htm
The above links are some interesting facts about saffron. Although they are not proper studies which prove anything, it might be just interesting to read them. Anyway as far as any beneficial effect goes it depends on the type of saffron, the country of origin, the purity, the method of extraction and the amount in the purest form absorbable by the body.
Padma's experience with saffron is pretty sad. My guess is some of the strands stuck to the back of the throat and irritated to trigger a cough. Even if she had an allergy to saffron it cannot last for so many months after she stopped taking it, so the cough intially triggered by saffron could've persisted due to other reasons. Anyways not to worry...
Just be a happy mom and am sure your baby will be happy too. Good wishes for a healthy baby.
-L.