Yuk. If you have to say 'hardly taste the difference' then you'd need to reconsider putting milk in your tea at all. I don't mind a vegetarian dish at all but please don't put fake food in front of me. Some say processed food is amongst the worst stuff to eat, so where does that put things like fake meat?
Have you actually tried good oat milks (not Oatly by the way)? Nothing like soya or other vegetarian milks. I wasn't convinced until I bought a bottle to try it out.
They're not quite dairy milk but the taste is definitely not unpleasant. I'd describe it a bit like a full fat milk with a bit less sweetness than cow milk. You can always add a sweetener or half a teaspoon of sugar if you wanted it sweeter. Of course, one benefit is it is almost as thick as full fat milk, but its calories are much lower than full fat milk per litre.
I'm not a vegetarian, but eating less meat and dairy is generally good for your health and the environment, and it's never been as convenient as it is now.
On processed food: I think there is a bigger hazard with meat with antibiotics and preservatives in the food supply for the animal, than say a mushroom or pea burger that has seasoning and binders added. If you
look at the ingredients in, say, a Beyond Burger, I don't think there's anything terrifying there. I don't think meat is that unhealthy in moderation, but I think you'd be kidding yourself to make such a generic statement as "processed food bad".