I think you’ve probably got over the biggest issue, which is realising there’s a problem, so that’s a really good start.
I was struck that you felt the advice from ilovemylampandchair was a bit extreme - it’s really not. We eat like that and always have. You can still eat like that and enjoy food, and still have treats sometimes - it’s not at all miserable! So:
Squash is a treat (probably only once a week, if that). Even fruit juice is once a day, no more - drinks are water, semi-skimmed milk, or DD seems to sometimes like herbal teas.
Breakfast: porridge, weetabix, shredded wheat, cornflakes, bran flakes. The DDs are happy to have most with no extra sugar. Teaspoon of honey in porridge. Sometimes granola. Pancakes and waffles with all sorts of lovely toppings (Nutella, jam etc) on a Sunday. But I still try to include fruit, eg bananas or blueberries, so that’s it’s not just 100% processed sugar!
Crisps are a treat - eg once a week at most. Usually not that often - we just don’t buy them very often.
Almost no fizzy drinks - they have them at parties if they want and we never buy them at home.
They are absolutely allowed cake, sweets, biscuits, but cake probably a couple of times a week, sweets less than that. At birthday parties they are allowed whatever they like.
Pudding after most weekday evening meals is fruit, or sometimes yoghurt. Often a pudding like apple crumble and custard etc on a Sunday.
How often are you having takeaways? Agree that’s another easy way of stopping high fat foods. Can you search on here for easy evening meals to have when you’d usually get a takeaway, eg baked potatoes, or chicken, veg and home made potatoes wedges (not chips) - just cut up potatoes into wedges, season, drizzle of olive oil and shove in the oven for 25mins.