Sainsburys can be on the more expensive side of supermarkets, but the good news is that tesco and asda offer a reasonably priced delivery service. I would seriously considering looking and see if they delivery to your area, I would imagine they do. £20 is definitely possibly but £25 is a lot easier and offers a lot more flexibility.
If you can get a delivery, I would include all your food for the first week, and then non perishable food only for the second week. If you keep this shop to about 35-40 pounds, then it leaves you an extra £5-15 for some fresh items from sainsburys for the following week.
I'm not going to try to give you an exact shopping list as that depends so much on your cooking ability and your personal tastes, but I'll just try to cover some general ideas.
A simple dinner boils down to a protein (chicken, beef mince etc), some carbs (rice, pasta, potatoes are great cheap options) and some vegetables.
Frozen vegetables are a life saver! So easy to prepare (boil or microwave them), and they are at least as good nutritionally as fresh. Not to mention so much cheaper, especially considering you'll probably need to throw anyway as they won't go off!
Carb wise, just shop the budget range, you won't notice a taste difference I promise. Rice and pasta will last a very long time so dont be afraid to buy in bulk if the price is good.
Meat wise (or you could go with some tofu if you like it) is definitely more expensive. So be prepared to eat the same meat throughout the week. I generally go with chicken, because it's just so versatile I could eat it for for breakfast lunch and chicken. But otherwise, sales are you friend. Get whatever is on sale, and if it's a really good sale, don't be afraid to freeze it. (Freezing is the secret to make your money and food go further, but I'm imagining freezer space will be limited based on how many people you're living with so just wait and see before buying a tonne of food you cant store!)
Now the optional (but not really) part is flavouring. Spices are a huge topic in itself but if I'm going to presume you're not a huge cook at the moment so I don't wanna overwhelm you. I recommend getting a very jars of a cheap sauce. Pasta and curry sauces are great (Never pay more than a pound for a jar! There is always a brand on sale!) The issue is you will not need a jar of sauce, so make you make the same or similar meal again within a few days so you don't waste it.
Leftovers are great so consider making slightly more food than you think you'll need for dinner and use it the next day.
Sandwiches are a staple boring choice but reasonably cheap depending on your choice of fillings. Some normal spreads like peanut butter and a piece of fruit make for a healthy lunch.
Honestly, you wont get cheaper than porridge, but it's not to everyone's taste. Get a store brand of sugary cereal (frosties cocopops, you know what I mean). It's delicious, it's cheap, you'll love it. It's not the healthy choice, but I promise you, the 'healthy' cereals are nearly all filled with an equal amount of sugar, and are nearly all contain the same amount of calories per 100g. But if your interested in being a little healthier, some fruit and weetabix is a great choice. You can use either milk or yoghurt. Yoghurts arguably healthier, and it's more filling, but it's more expensive.
Fruit is a great snack. Bit of a fruit with yoghurt, so healthy and delicious and the calories are minimal. Dont be afraid of tinned or frozen fruit! The tinned stuff generally is tinned in a syrup so the calorie count is higher but it's still a much better choice than a chocolate biscuit so dont stress about it.
If you're anything like me, you're gonna crave some biscuits or some crisps at some point. And if you don't buy the cheap store brand in your shop, your gonna walk round and pay 3 times the price at the local corner store. So dont be afraid to get a cheap packet of biscuits or a 12 pack of crisps. Just don't over do it, when they run out, don't run off to get more! Just wait till your next shop.
Sorry for the long post! Well done if you've got to the bottom of it, hopefully that wasn't too much information. Don't take any of what I said as gospel! It's just what I've learned from trying to eat a balanced diet as a student, good luck and have a great time at Uni!