If you don't know French and wanted to be less likely to mess anything up with what you were saying, you could also maybe say something simple like,
"No nuts, please! I am allergic to them!" = "Aucunes noix, s'il vous plaît! Je suis allergique à eux!"
Or maybe, "No food with any nuts in it please." = "Aucune nourriture avec n'importe quelles noix dans lui s'il vous plaît."
Or even, "I am allergic to nuts." = "Je suis allergique aux noix."
As in the first answer given, all of the translations above are also from
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I know what you mean about a nut allergy being very dangerous! I have gone into anaphylactic shock on one occasion because of eating a nut. It was the worst physical experience of my life!