Cutting out meat and dairy will help. Dairy free does not need to be expensive, if you are then buying dairy free cheese and yogurts, these can be expensive but just the same as dairy. Dairy free ‘butter’ is pretty cheap and Lidl sells one litres of soy milk for 59p.
We are on a smallish budget, a family of five for around £60 a week. We eat things like Dahl, sweet potato and veg curry, chili, homemade pizza, homemade veggie pies and pasties, baked tortellini, toast and beans, veggie sausages with mash and veggies, Cajun rice and sweet potato bowls, vegan chili Mac and cheese, veggie fried rice and curry sauce, bean and veggie enchiladas, baked sandwiches and wedges, homemade soups, twice baked potatoes (freeze them in big batches and then microwave when needed) healthy muffins (they have hardly any sugar and lots of oats and frozen fruit! Great for breakfasts), simple cookies, protein balls, porridge, pancakes and homemade fruit sauces, etc.
We eat hardly any meat or dairy, instead opting for different beans (kidney, black, chickpeas, pinto, etc), lentils, yellow spilt peas and veggies. I usually only get meat when it’s on a really good offer or reduced. I batch cook and freeze when and what I can or even just making one batch of breakfast muffins will last for a few days (we normally get two days as bake in 12s so with five of us that’s all we get lol, with only three of you you would get more days obviously)