Green Cuisine really doesn't deserve to be described as "just" veg-friendly as the non-vegetarian stuff (fish) features so low. What you get here is one of the nicest vegi/vegan cafes I've ever been too. Everything is cooked on the premises from locally sourced organic produce, and the main chef and owner is a fantastic cook. There is a good range to choose from, ranging from soups and salads to good sized meals. There are toys and books to keep kids entertained whilst you wait and it's licenced as well. Last Sunday of the month the cafe is opened for roast dinners (kids eat half price) and there are evening events as well. To finish off your visit there is usually a good selection of vegan and gluten free cakes and puddings to choose from.
It's a bit off the beaten track if your're shopping in Worthing, but well worth the detour. I go there regularly and I'm never disappointed. I would give it 5 cows but I'm not allowed to as they serve fish.
Cons: Can be a bit pricey for a family visit, A bit of walk to get there