Oh I do love a challenge 
I don't have kids but do live on a very very tight budget for food. I'm also trying to lose weight (lost nearly 2st since Christmas!!) so we eat very healthily if we can. These are a few of the things which really help me:
Aldi's super 6: 25p for a range of different veggies on offer every week, so for example 25p for 2.5kgs of potatoes this week! Also 25p for a kg of carrots. The veggies change every week but potatoes are on the list pretty reliably and so are carrots. Carrot sticks make great snacks for kids and adults and are cheaper than crisps!
Buy joints of meat and butcher them up into portions. So for example a £5 pork joint will make between 10 and 15 thin pork chops and the ends will make a pork casserole.
Swap chicken for turkey and don't tell the kids, they cant taste the difference and it's butt loads cheaper!
Again, you can pick up frozen turkey crowns on offer for £7 and they'll do about 12 portions.
Frozen fish portions: Iceland or Morrisons do these and I think you get 6 in a pack. 1 will do 2 kids easily and they are £1.70 a pack.
Frozen sausages: 20 in a pack for £1.92 in morrisons. 2 each for adults and 1 each for kids so they will do you 2 meals. Bulk out some mashed potato with mashed carrot or swede to make it go further for cheaper and it's healthier too!
Buy a slow cooker immediately. £14 in Morrisons and mine saves me a small fortune. You can buy dirt cheap cuts of meat, bung them in the slow cooker to stew all day with loads of bulky veggies and you've got a home cooked meal when you get in for next to nothing that will serve 12. We make 12 portions of stew at a time and freeze it into portions.
Learn how to make very quick short crust pastry, half fat (low fat marg) to flour, Left over stew becomes pie for next to nothing.
Dont waste anything! Turn all veggies into mystery soup - in our house we dont buy any veggies that cant go into a soup and that soup then becomes a starter for a meal which makes you eat less meat and expensive stuff for your main :)
Eggs eggs and more eggs. Omelettes packed out with a ton of veg provides protein without the price tag of meat!
For snacks, apples are really cheap and sweet!
Rice is another good way to bulk out meals, standard white rice will make "risotto" - essentially tinned tomatoes, loads of veg and mushrooms all stewed down until it's thick :) It's the same flavors as a pasta meal but much cheaper!
I have a ton more, I feed 2 adults for £14 a week at the moment and we dont go hungry! 