My DD is egg free, milk and soya intolerant and we really struggle pudding front when we are out but at home have found enough to keep us going.
She can have soya as a minor ingredient which is a blessing but not in the form of soya milk, yoghurt, cheese, butter, ice cream etc.
She is ok with baked in milk, so fine with biscuits often, just watch for egg sneaking in.
At home for puddings we have:
Fruit with honey and sprinkles on top (check the sprinkles)
Jelly
Decorate your own biscuit - Rich tea biscuit with Betty Crocker buttercream style frosting (egg/dairy free) and sprinkles all on a plate and let them decorate and then eat - their favourite
Christmas pudding -stock up at Christmas, found little individual egg free, dairy free, soya free ones in Sainsburys, the individual be good to yourself range. We have them all year 
Cake - basic cake mix, made up either with water or milk substitute (almond works well). I make a batch using this one, for 22p, and freeze half. I've had a chocolate mix too that was egg/dairy free too but can't remember where from now
Soreen Malt Loaf (Original) with Pure Sunflower spread on top
Homemade air-popped popcorn
Fruit flavoured ice lollies
Sorbet - we always just double check for egg white lurking on the ingredients
EcoMill Almond chocolate pudding which they also do in Vanilla
Wot,No Dairy? yoghurt, they do a few flavours
Ogran pancakes made up following the vegan instructions, we use almond milk or water or half and half.
Fruit pots
And when inspiration fails we have biscuits.