'with the exception of milk & cheese I would [be] vegan'
You do know Quorn is made with eggs, right? (and it's only a few yrs ago it stopped being battery eggs).
That minor quibble aside, doubt being vegan per se is making you ill - my house has 3 vegans & i run with another, and I don't recognise any of what you are describing.
I would suggest trying a basic cheap multivit for a few months, on the grounds that it can't do much harm. Try & get one with iron, 'cos leafy green veg isn't actually as good a source as is often thought (oxalacetate inhibits absorption).
The
Vegan Society has more info than you will ever need on optimum nutrition & plant-based diets.
Mainlining milk & cheese as your main protein source has several disadvantages - you are probably getting more saturated animal fat than a meat-eater, and if you have a dairy intolerance (sore throat & snotty nose could be that) then obviously it won't help.
You also don't seem to have obvious sources of 'good' fats in your typical day's eating. I'd suggest a handful of nuts & seeds, especially brazil nuts (selinium, for immune system), pumpkin seeds (zinc, omega-3), & walnuts (omega-3). Flax seed oil is another useful source of omega-3 (NOT whole flax seeds, they are just a laxative), but it tastes icky.