How common is this in other cultures? I can see how it could have taken hold in the UK in previous generations because you wouldn't stand out that much if what you ended up eating was:
Porridge, cereal, toast, eggs, bacon etc for breakfast
Cheese or ham or similar sandwich at midday, no salad, no fruit - maybe a dab of pickle, which doesn't really count as veg - or if you're in a work canteen or caff and have a hot meal, pie and chips, egg and chips, shepherd's pie etc
Bread and jam, cakes, biscuits with tea at the end of the working day
Sunday dinner: roast meat, potatoes in some form, gravy
(Bit trickier if the veg refuser doesn't have potatoes, and that would probably mean a big drop in Vitamin C intake)
Far harder to see how you could avoid vegetables in other traditional cuisines.