My son can't have dairy or soya and his first birthday cake was a bought honey cake made by Solebio. It does have rye in it, but you could probably use a honey cake recipe and substitute gluten-free flour for the rye.
You might be interested in The Child Friendly Allergy Cookbook because her 'Wonder Cake' is free of all the stuff you're avoiding - but I have to admit that it turned out quite crumbly when I made it!
I also have an apple cake recipe which is our standard at home. I've made it with normal flour or gluten free flour and it doesn't seem to matter. I usually put eggs in it but it might work okay with egg replacer. Can't locate the recipe at the moment becase I copied it out for a friend and now it's not in my recipe file. But I can search it out if you like?