My DS has the same.
At 16 months now can tolerate egg in baked goods and soya lecithin. So hopefully we are getting there. We have been told to wait till he is 2 to introduce milk and wheat.
In terms of food, Ella's pouches, most of the fruit and veg based ones are fine. Also organix rice cakes and sweet corn crisps and carrot sticks are fine. Other snacks we use alot is nakd bars (full of nuts and dates), raisins, other dried fruit, and nairins oat biscuits.
In terms of food, I tend to make alternate between rice and gluten free pasta, alongside protein, so chicken, lamb, or fish (and veg of course). Tesco do nice free from fish fingers, and mccains homefries I think have no wheat in them. Tin of beans are fine too, so is mash, and I think so are waffles, though DS isn't too keen on them.
In terms of pudding, there is jelly, and you could bake cake with gluten free flour. I've also milk made milk jelly with coconut.
Sadly alot of the gluten free stuff in the shops have egg in them too. Have you tried egg in baked goods with your little one? Warburtons Newhouse do a lovely gluten and dairy free lemon and poppy seed cake (in the gluten free section in the shops). My DS practically lives on nakd bars though.
We are lucky that GP has agreed to prescribe gluten free stuff for DS, after I asked dietician to write to GP. So we get juvela gluten free flakes which are high in iron, and DS has with coconut milk. Also I've recently started giving ready brek with coconut milk, as it has ALL the calcium they need (it does have a not recommend for milk allergy sufferes note but after looking into online I decide to risk, and my DS didn't react to it).
I also give multivitamins and he is on vitamin D supplements too and is meant to be on iron supplements too but doesn't like them 
Also, we get warburtons gluten free bread (sorry has egg in it) on prescription, and that has added iron and calcium too, so I usually spread on it the free from hazelnut and chocolate spread that you can get from the free from section in a large sainsburys. This is soya free too, and honestly it is the yummiest gluten free and dairy, soya and egg free chocolate (and trust me, I've tried loads!).
Hope that helps, if you have any questions let me know 