Oh dear, threads like this make me want to weep (not because of you though OP). You can have an anaphylactic reaction to ANYTHING. Nuts are NOT the only food that can kill you.
My DD (7) has a severe allergy to milk. A couple of years ago whilst on holiday she had a bread roll for breakfast which we realised after had milk as an ingredient. After a few minutes, she started coughing and then coughing so badly that she was practically retching. I thought she was going to be sick but no, this was was just a sign that her airways were closing up. She then got very distressed, said she had tummy ache and needed a poo (another classic sign of an anaphylactic reaction) and then, worst thing of all I could see her chest pulling in tight as she was struggling to breathe. We hadn't been prescribed an epipen at this point but luckily a huge dose of piriton brought her breathing under control. She was still left for some time after with swollen eyes and lips and a very sore throat.
Is this a suitably bad reaction to something other than nuts for all you disbelievers? It was certainly a far worse reaction than my neighbours DD ever had to nuts and yet her mum forced the school into having a nut ban.
So DD goes to school where nuts are banned. It's mentioned regularly in the newsletters and on the website. But its also a school where despite knowing that a child could have an anaphylactic reaction to milk products, we've had class/year parties where parents have been asked to bring in pizzas, cheese sandwiches, lumps of cheese.....
On school trips (her year only), they are told over and over, bring a packed lunch but NO nut products. There is no one in her year with a nut allergy!
So I would say to you op, don't use a nut ban as a reason for chosing a school. They aren't enforceable, and from my experience don't give me any confidence that the school really understands food allergies. To me they are just paying lip service to something because they feel they should do.
Instead I would continue to reinforce the belief in your dc that the world cannot be a guaranteed safe place with regards to food and that they shouldn't swap food with friends or eat something without knowing what's in it.
Sorry for the epic post but nut bans do get my goat. They aren't enforceable and just continue to put out the belief that only nut allergies are bad