Do YOU want to go vegan? If you don't then he is free to cook vegan when it's his turn to cook and do a batch so that there is plenty in the freezer when the rest of the family are having an omni or veggie meal.
I can say that when DP and I went vegan we talked about it for a couple of months before making any decisions and we shared the research and cooking. I think if we hadn't then one of us would have ended up resentful.
If you do want to try veganism in principle, but you're worried about the issues in your OP, then here are my suggestions
first and foremost my tea. I drink milk in my tea. We tried oat milk a while back and even he decided it wasn't great. Is there a better option?
Alpro vanilla is quite pleasant but it won't be the same. Koko unsweetened is also nice.
Mushy food makes me gag. He's talking eating lentils, peas and beans. Is there any way to prepare them so they're not mushy?
Yes. It depends on the bean or lentil. Red lentils will always go mushy whereas beluga lentils won't. Likewise black eyed beans are intended to be mushed down, while chickpeas and red kidney beans will keep their texture. Assuming they're tinned, don't cook them for too long!
We, especially DS, eat a lot of cheese and yoghurt. How do you keep a growing teenager full without cereal/milk/cheese on toast? We're not looking to try out highly processed vegan copies of non- vegan food.
Peanut butter!! Peanut butter on toast, apple slices with peanut butter. If peanuts are out due to the nut issue you mention then you can get sunflower seed butter from health food shops. Also tahini on everything. Avocados. Cereal and soy milk.
I cannot eat mushrooms. Well, I can eat them but then get horrific stomach cramps a few hours later. Lots of recipes seem to be mushroom based.
Substitute tomatoes or celery?
How do I make sure the DC are getting all the nutrients they need as they come into and through puberty?
There's good advice on this on the Vegan Society website.
Nuts - hazelnuts make DC2's mouth blister, walnuts make my mouth itch and I don't like any other nuts.
Get Dreena Burton's recipe book Plant Powered Families as she gives nut free alternatives to her recipes. I've used that book for years and every recipe is fantastic. Lots of non-mushy food there too!