You might get more advice if I move this to the allergies forum - let me know if you want me too.
YOu might also want to do a search on birthday cake in this forum as I'm pretty sure this has been asked before.
Here's one of my recipes that has been enjoyed by many of my friends and relatives - for non-dairy version, use soy yogurt or a little less non-dairy milk with a tablespoon of vinegar for the yogurt.
Gluten-Free Bundt Cake with Apricot-Orange Glaze
This cake is so moist and delicious, no one will guess it has no wheat or eggs in it. It is good on its own or with the glaze.
1/2 cup very soft unsalted butter or coconut oil
3/4 cup maple syrup, brown rice syrup, agave nectar, or honey
2 tablespoons flaxseeds, ground
1/4 cup water
1/4 cup orange juice
Grated peel of 1 orange
1 cup plain yogurt (dairy or nondairy)
2 teaspoons vanilla extract
2 1/2 cups brown rice flour
1 teaspoon baking powder
1/2 teaspoon baking soda
1/4 teaspoon sea salt
1 teaspoon ground cinnamon
1 teaspoon ground cardamom
1/4 teaspoon ground nutmeg
Apricot-Orange Glaze:
1/2 cup 20-Minute Apricot Jam (page xxx) or prepared apricot jam or preserves
3 to 4 tablespoons orange juice
Preheat oven to 350ºF. Oil a bundt pan. Beat butter or oil and sweetener together until smooth. In small bowl, mix ground flax and water. Add to butter mixture along with orange juice and peel, yogurt, and vanilla. Beat well. In separate bowl, sift remaining ingredients together. Stir liquid ingredients into flour mixture. Pour into prepared pan and bake 45 minutes, or until tester inserted in cake comes out dry. Let cool 10 minutes, then remove cake from pan.
Place glaze ingredients in saucepan. Stir together over low heat until warm and thinned out. Spoon glaze over cake if desired. Serve warm or room temperature.
Makes 12 servings
I didn't think of that. If you think it would help, go ahead and move it. And duh again, "cake." I did a search for "gf/cf." 
