Ignoring other costs, £20/week is absolutely doable. Across your first few weeks, stock up slowly on bases and basic things like pasta (40-60p), rice and couscous (£1-2), oats (50-70p), pasta sauce (usually a few portions' worth for 50p-£2), dry noodles (£1-2) and tinned beans/tomatoes/chickpeas/kidney beans/other tinned foods like tuna (anywhere 10p-£1). Keeping things like this in your cupboard is super important since it lets you have simple, easy-to-make meals at any time - and if you get these over the course of a few grocery shops, you're barely spending anything.
Frozen fruit and veg are also some of the greatest things ever for students. Grab a frozen stir fry mix, something like a broccoli/cauliflower/carrot mix, and some peas and/or sweetcorn, and you'll be sorted for a pretty decent amount of time. I buy my frozen and cupboard things probably not even once a month and they last me well. Week-to-week, you only need to stock up on any fresh dairy, fruit/veg (which you won't need so much if you have frozen), and flavourings and such. I tend to just buy bananas, oranges, potatoes, milk, and anything meal-specific throughout the week.