Chocolate is always a safe bet - nice and messy. Or a Victoria sponge. Apart from nuts I gave mine pretty much anything from one (and I was more worried about choking than allergies) I assume the child in question has already had eggs/dairy etc?
Oh dear - it's getting technical here. I think eggs is 6 months, unless there is a history of allergies. I gave yolk and white at the same time - the only advice I was given was that the egg had to be cooked so well it would bounce off the floor. I would make sure eggs are not a problem before giving a cake with eggs in. Cream should be fine as she has had milk and yoghurt.
But I'm not an expert - have no experience of allergies and so I haven't ever been careful or wary of introducing new foods.
It is yolk first (used to mash into potato for dd). A plain victoria sponge or madeira cake is good for a first birthday - or you could do fairy cakes and lay them out in the shape of a 1 or decorate with the letters of her name .