This is a nightmare for parents, and they worry a lot, and I am not surprised. Any school at all should be able to put things in place though. Legally they are required to, as long as the restrictions you ask for are coming from a reliable medical professional.
I have worked in many state schools, mainstream and special schools, and in most schools there are children who have epipens. As it happens, not in my current one! Children have a host of different needs and diagnosis now, which are often controversial, contradictory and confusing, However, anaphylactic allergies, and diabetes, are both in a class apart from all other SEND. Lists and photos of affected children are widely available to all staff at all times, all staff know where the pens/ treatments are kept, and who is trained to administer it. In some special schools, all staff are trained to administer epipens.
Nuts are banned from my current school, even though no child or adult present is known to have a nut allergy.
In cases where severe allergies are known, rooms and areas and procedures are put in place to keep the individual safe
It has been drummed into me so hard that I don't eat peanut butter even at home during term time! It is a holiday treat only.
( and I am not being rude or doubting you saying information has to come from a reliable medical professional, we have in the past been asked to make adjustments instructed by some very weird alternative practitioners ( a later start on days when the weather is changing - school bus to not go above a certain speed - child's desk to always face East, of course, those infernal Iren syndrome doodas, and so on........ and those were ignored)