It really depends on the person and how sensitive they are. I'm lactose intolerant and I usually know an hour after I have something if someone put so much as butter on it. If someone mixes up my coffee I know in 10 minutes because the bloating gets so bad.
I'd also imagine food allergies don't act that much differently from other allergies- I'm allergic to fragrance but it could take a couple hours or days for me to experience symptoms. I'm also extremely allergic to lavender- it would take minutes for my skin to swell up and less than 2 hours for me to get hives.
So it would really depend on two things: how allergic that person is to a specific thing and how much of the allergen is ingested. Also not all allergies and intolerance shows up with diarrhea- you could also get bloating, rashes, swelling, nausea, headaches, etc.
If you think you're experiencing an allergic reaction to something, don't have it again (if you know what it is). You can also talk to the dr who can help you find out what you may be allergic to and depending on the amount of reaction- how much you are allergic to it.
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