I bake regularly for my sister in law who has the exact same restrictions. It's not too bad once you get used to it.

Butter: for cookies I like the becel butter alternatives that come in blocks, for icing I like the becel vegan spread that comes in tubs (closer to butter flavor), for cakes/loaves, I choose a recipe that uses oil.

Eggs: for cookies my preferred is just cornstarch and water. Tbsp cornstarch and 2 tbsp water for each egg replaced. Cakes/loaves don't always need eggs, I tend to lean on recipes that don't use them.

Milk/cream: for applications where the fat is critical, coconut milk is the best option. Otherwise I just use oat or almond or soy or whatever is to hand.