I run a community group in my area baking birthday cakes for children whose families can't provide one (it's called Free Cakes for Kids - I first read about it on here, during Mumsnet Giving Week).
We've been asked to bake a birthday cake for a child who is allergic to eggs, nuts, shellfish and fish. There's no way this child will get a birthday cake otherwise and I would love to be able to provide one for him, but I don't think any of my team's kitchens would be completely free of these allergens.
We don't bake with nuts and would use baking tins that hadn't had nuts in them; we can also use an egg free recipe. If your child has similar allergies, would you be comfortable with a cake being baked for them in a kitchen which has been scrupulously cleaned, but which has had the allergens present in the past. The cake definitely won't come into contact with any of the allergens, but I would hate to put the child at risk in any way.
(My children all have coeliac disease, so I'm not unfamiliar with cooking for special diets, but at least in my children's case, if they did eat gluten by mistake, there wouldn't be an immediate life-threatening reaction, so I really don't want to risk anything.)