But you said no alcohol. Does anyone at the picnic drink? Could you take Ribena for the teetotalers and Pimms for the fun folk? 
Nope.Well, my generation drinks, but we don't drink at the picnic because my uncle (who owns the farm) is against drinking. And they might even live in a dry township, now that I think about it.
Pasties are easy, they're just little pies. Make some pie crust, or buy some ready-made, cut it into circles about 6 inches in diameter, fill them with steak and onions or chicken and veg or whatever you want, really, fold them over to make a little half-moon shape, seal the edges, brush them with egg and bake until they're brown. I think proper cornish pasties have a hole poked in the center, but I'm not certain. I'll find a recipe for you if you're interested.
So, sort of like a mini italian stromboli? I can handle that! I made mini stromboli by taking a tube of pillsbury biscuits, flattening the biscuit, putting pieces of string cheese and ham inside, folding it over and baking. Sounds like I could do the same thing minus the cheese to make a pastie.
Or how about cucumber sandwiches?
I thought about doing that, but thought that was more of an afternoon tea kind of sandwich? Although, I suppose since it's a sandwich, it's a picnic food!
Cheese is dairy though right? 
The cheese has a better chance of surviving the cooler and kiddie-pool with ice though over something that has say, whipped cream in it, I think.