You say that you'd rather shop in primark for your children, regardless of your financial situation but what happens when they are 15/16 and they want Michael Kors not Primark? Are you going to say no, even though it's within your budget?
DD is 16, DS is 14.
DD loves fashion but more likes things for what they look like, rather than the brand, if that makes sense. She has expensive clothes and shoes and things, but not because they're expensive, because she likes them.
DD's school bag is from Primark
It's lovely - she's got lots of compliments on it. IME it's not actually high-income families that always have the expensive branded accessories.
They all wear uniform but generally if we're going clothes shopping the price won't matter too much if they like it - within reason of course. I don't think I'd pay £1000 for a coat that they were taking to school - it would get lost in a week knowing them!
I think by secondary school age they'd definitely connect that someone rich will be able to have expensive stuff. If they've been to Lauren's house, and they know she went to private school, and they've seen her dad's car etc. etc. I daresay they will work out, whether or not she's allowed to bring it in, that she could afford and probably has designer clothes.
It's sad, I guess, but it's how the world works unfortunately. I think there comes a certain age when we need to stop "protecting" our kids from inequality - when it's something they can see anyway - and teach them how to act in the face of it.