"Mon enfant est allergique aux arachides (noix, noisettes, cacahuetes, huiles..). Cette allergie est tres grave, et peut entrainer la mort. Est-ce que vos produits contiennent des arachides ou derives? "
Sorry, I don't know how to make the accents on a laptop.
That's what I would say. Arachides is the broad term for nuts, it includes all types of nuts (peanuts, pecan nuts, walnuts...). I'm French, btw, if that makes you feel more confdent about using my version.