Bullshit.
I'll preface what I'm going to say next by saying I'm in the US and I know this was a UK report - but I highly doubt things are so incredibly different over there in terms of cost. That being said, there's utterly no way that has to be the case. Raw produce (not meats) from a farmer's market or most major supermarkets are very cheap. I can spend $30-$40 and have enough for 3-4 people for a full week. Also, sausage is usually very cheap. I buy the store brand (which happens to be delicious) and it's usually <$3 per lb. The 73% beef from Walmart is cheap as hell and usually hovers around $3-$3.25 per lb most of the time. One of those burgers (quarter-pounder, so 4 oz) has 340 calories and is a dream for anyone doing keto. So that $3 package contains about 1350 to 1400 calories! The cereal I make myself has only 4 ingredients and is loaded with healthy carbs (steel cut oats, flax seed, walnuts, and raisins). $12 to $15 dollars buys a bag of each of those ingredients which all have around 15 or so servings - that's over two weeks...or less than a dollar a day (per person). And that is a very high calorie breakfast, BTW. Water costs essentially nothing per gallon. Eggs are cheap and super healthy. There's plenty of cheap, tasty cheeses you can buy in bulk, too. You can get whole seasoned, rotisserie, cooked and ready-to-eat chickens for $5! That's multiple meals for a single person or a meal for an entire family. And I don't live in a rural part of the US where prices are cheaper - I live in an expensive area of the northeast.
You can easily feed a family of 4 on $125 a week using the items I outlined above (along with other cheap options).