I am sorry, I am of no help because my dd had a chocolate cake for her first birthday!
But I have heard of alot of moms doing a carrot cake for her first bday. What about an angelfood cake with fresh fruit on top? I dont think it has eggs, wheat or milk, but definately sugar. If you did a homemade recipe you could substitue the sugar for concentrated frozen apple juice. I have done that in cookies and it turns out great!
Good luck, I bet other people will be way more helpful than me.

But I have heard of alot of moms doing a carrot cake for her first bday. What about an angelfood cake with fresh fruit on top? I dont think it has eggs, wheat or milk, but definately sugar. If you did a homemade recipe you could substitue the sugar for concentrated frozen apple juice. I have done that in cookies and it turns out great!
Good luck, I bet other people will be way more helpful than me.