I'm not vegan, but I have tree nut (with cashew as prime), mango and dairy allergies, so I do eat a lot of vegan tree nut free food. And in Lent, that's my Fridays.
Olive oil, Melt vegan buttery spread, and canola oil are my go to fats. The 1st is what I do almost all my savory cooking with. The 2nd is what I use when I need a "buttery" taste. And the last is what I tend to bake with (although I also use Melt, but Melt is pricey and money is an object for me). I also keep coconut oil in the fridge for some baking my daughter does for me.
For milks, I use oatmilk, rice milk, and Asian canned coconut milk. Oatmilk is for my coffee/cereal (I don't drink cups of it), rice milk is for mashed potatoes (with olive oil and Melt) and savory/sweet cooking where you don't want a "flavor", but just a sweetish liquid, Asian canned coconut milk is for my Asian savory cooking and my sweet cooking where a coconut flavor would not be off-putting.
For proteins, I use fortified nutritional yeast (I make a vegan "alfredo sauce" - and this product delivers so many B vitamins and proteins, it's great), peanut butter/peanuts, tofu, quinoa, lentils, and beans (black bean and great Northern beans are my favorites). I don't buy processed "plant based" proteins b/c I don't want overly processed foods (and as you've found, most have hidden tree nuts), whether vegan or not. So, I don't do the cheeses or the meats made of veg, unless it actually wants to be the veg. So, my one "plant based" protein is a black bean burger, b/c that has always wanted to be black beans
. I also do portobello burgers.