I have several of JO's books, and actually do use them! Jamies Dinners and also Jamie Italy are actually quite good. Don't like the 30 min one or the Great Britain one muc personally.
I think it would be pretty boring either watching programme about cooking on a £40 for a family of 4 budget or even buying the book. Realistically your options are limited. Pasta (dried), seasonal veg and cheap cuts of meat will be your friend. As will rice and pulses. And I wouldn't pay £15 for a book to cook like that, nor would I watch a TV programme for 30 mins that had recipies that use those ingredients.
I have cooked on a budget before. Me and DD and DP on about £30 a week for 6 weeks. It happens. We managed. Wouldn't want to do it again.
Now if I take the time and effort to follow a recipe I want it to be something nice. It doesn't have to cost the earth, it doesn't have to be WOW, it just has to be something tasty. Can't afford to spend a fortune. Lamb at full price I don't buy as it's too expensive, as is cod but give me either half price or better and I'm in. See a chicken roast as a good value meal and we have the odd takeaway. To give you some context of budget.
When he did the Ministry of Food I worked in the town it was being filmed in, on a butchers market stall. We were told by the producers that the programme was what people have suggested. But the budget was higher. Family of 4 being fed for a week on £50. But when he started researching families to work with many didn't have the skills to be able to cope, so it changed to the Ministry. According to what we were told.
Interestingly all the grub he used making that show was bought (AFAIK) from the local market and shops. Good for the local economy at the time. However Rotherham Market (though a good market) isn't known for its ethically sourced/free range/organic approach to produce. In fact, I can't think of one butchers or veg stall that has that sort of stuff as people just wouldn't pay for it!
He has done good things. School dinners in particular he raised lots of issues and has made a change for the better. He won't get it right all of the time but I think cooking on a real budget just would be good telly, nor would it make a good book. There are things and meals we all know that we use when its 'skint week', maybe we ought to get a collection of them together and release them as a book?