Oh, my heart breaks reading both of your messages.
So what I am about to say next is hard because I don't want you to think I'm telling you that you're bad mothers, but the only reason the kids eat what they do is because it's offered and allowed to be their diets.
I'm a full-time working mom with a 2 year-old and an almost 4 year-old, and I completely understand time constraints. I'm also a recent cancer survivor and the wife to a husband who was allowed to eat poorly his whole life and is ridiculously picky.
Our kids really don't have a choice - sure, we'll give them a few options of things we're considering eating and let them tell us what they're in the mood for. We have chicken nuggets in the freezer, but they're a once/month kind of food vs. once/week.
We always have a fruit or vegetable on the table. Fresh is best in most cases, followed by frozen, canned in their own juices, and canned in syrup. We almost exclusively do fresh or frozen.
Having spent almost a decade in the world of Diabetes, having a mother with type 2 diabetes, being genetically predisposed myself, and having a 35 year-old husband on both heart/cholesterol medication, I worry about our kids as their generation is the first expected not to outlive their parents.
I'd recommend keeping a few staples in the house (we get most of ours at Costco to keep costs down) including chicken breast (we freeze until we're ready to dethaw), ground turkey (tacos, turkey burgers, meatloaf), eggs (we do a lot of breakfasts with fruit for dinner).
If you can find a few veggies they like (it may take lots of coaxing), I'd recommend adding those to your weekly shopping list. For us, it's carrots, broccoli, cucumbers and recently celery and green/red peppers.
We do a lot of quesadillas/fajitas/tacos because they're easy to make and relatively inexpensive. We use either low-fat refried beans or low-sodium black beans for protein if we're not having meet and make sure lettuce/cheese/tomatoes are in the tortillas.
If we're doing pasta, we keep the quantities limited and always serve with a fresh vegetable lightly sauteed on the side (often quicker than steaming frozen vegetables).
Turkey burgers with a L. bit of ranch dressing for dipping with carrots/cucumbers on the side is a favorite in our house.
Jamie Oliver gave a great recipe for homemade chicken nuggets on Oprah to promote his Food Revolution show on ABC - I'm sure the recipes are available on either Oprah.com or on the Food Revolution site.
Be careful of any prepackaged foods because they're usually really high in sodium/sugar for preservation and flavor.
Check labels to compare turkey-bacon to pork-bacon. You may be surprised at what you find.
Finally, you have to set a good example yourself. My husband has learned to tolerate broccoli in the best interest of his own health and the kids' well-being. Hiding veggies in foods only encourages them NOT to learn to like the taste of them, so I'm really opposed to that approach.
In the end, you're the parent, they're the child. Sometimes being the parent and making them upset is hard, but if they're best interests are first, you can't go wrong.