I cook LC meals. My SO has the option to eat them or fend for himself, he usually eats at work anyway.
For my 4 year old I usually cook him what ever I'm having unless it is something spicy, or shashimi or other thing I would worry about him eating. Basically I will either make him a small side dish, or if he eats a decent portion of the LC meal he can have something else after.
Tonight we had spaghetti squash with marinara sauce and turkey/lamb meatballs. I thought he would eat the spaghetti sauce and nothing else. I did put a small meatball on his plate and asked him to at least try it, though he didn't have to eat it. He ended up loving the meatballs and eating 4.5 of them
Suprised the heck out of me. He didn't really eat the squash though. As a treat for eating well at dinner he got a banana, some sesame pretzals, and a little diet soda for a treat.
Last night we had salad. I made his with ranch dressing and a little chicken breast. When he ate at least some (the veggies are a fight) he got a treat of "chocolate" yogurt. Basicaly it's whole milk plain yogurt with some protien shake mixed in for taste. He's happy because he gets something special (don't tell him it could be healthy) and I'm happy he's at least taking a few bites of what everyone else is having.
Atkins is high fat though, and mixing high fat, plus high carbs for the family can be very bad. When she makes meals, she can add oils to hers for taste, add cream to snacks and so on to get her fat intake up, while keeping the meals for the rest of the family moderate fat. With children that young they need the fat for brain development anyway. They don't need sugar. Giving them more of a balance of moderate carb/fat instead of low one or the other would be best. When they see her eating healthy and there isn't junk food in the house, they will eat healthy too by example. For the kids she can also reserve out a small portion of the meat she cooks before she adds in any spices, or make things with two seperate sauces.
Make sure she keeps plenty of fruit around for the kids to snack on. After induciton when she can add back in things such as whole milk yogurt, berries, cottage cheese and so on, the whole family can enjoy these healthy snacks. String cheese is also a wonderful LC treat children love.