managing food on a low budget and busy schedule

For us:
Chopped up fruits and veggies. Picking 2-3 days a week to just spend a few minutes chopping up some veggies/fruits and storing them in special containers in the fridge for ready-made snacks was a nice one. The bins are always the same ones so the kids know where to look for them.

Nuts - plain nuts, you can take a little time over the weekend and make your own flavored nuts. Anything from savory to sweet, roasted in the oven for just a few minutes, and then stored for kids to eat later.

Roasted chickpeas - you can buy dried chickpeas, cook them, and then make roasted chickpeas the same as you would flavored, roast nuts. Although these tend to be much cheaper, only taking some time (we would do over the weekend while, say, cleaning house or doing other chores). Many recipes online for varieties of roasted chickpeas, and pretty healthy. We've tried it with other beans and they don't taste as good, but some are doable.

Eggs - if they like eggs, hard boiled eggs are easy, and you could maybe do a type of deviled eggs without too much effort, to store for a couple days here and there.

I buy at the market not at the supermarket and that saves a lot of money. I have two kids and a partner also and we eat on a budget. I find filling most of the plate with veg or salad a lot cheaper, also buying in bulk and making your own from scratch rather than pre-packaged foods saves money also.

I personally take one day a week (normally Sunday) and make all my big meals (i.e. Lunches ) for the whole week as well as some snacks for the afternoons. I buy most of my fresh ingredients at the supermarket in the morning and make the meals as soon as I get home. Normally it takes about 2-3 hours to make 5 meals but you do have to have a meal plan for the whole week. Honestly, I try to make meals that use similar ingredients to reduce my cutting time and cut everything first before turning on the stove. Once everything is cut and ready, it goes pretty quickly. Also, I think it would be pretty easy to incorporate the help of your kids in the prep time.

As for good snacks for kids:
Appels with PB.
celery with cream cheese
cheese balls (100g guyere cheese, oil and an egg, have the kids help. Mix the egg and the cheese together, roll them into little ballsand fry them at high temp until the outside is crunchy. They are also good cold.)
Carrot muffins (http://allrecipes.com/recipe/awesome-carrot-muffins/)
Homemade granola with yoghurt (mix all your favorite ingredients : oats, rasiens, nuts, chocolate chips, seeds, add a little bit of maple syrup (or brown sugar with water works as well) and pop into the oven for 10 to 20 minutes until golden. It keeps for about a week in an airtight container.)