I'm from a very poor background but was lucky enough to just scrape through university before full maintenance grants were removed. So much of what people have already said resonates; I'm now fairly well educated and not afraid of news sources like the Guardian but when I was growing up my dad kept telling me it was much too highbrow for the likes of us. And ironically he was the parent who desperately wanted me to get an education! There is no way I'd have been able to go to university now - 27k of debt minimum just for your tuition? That's way more than many people earn in a year in the background I'm from - and it's not that they don't work hard, despite what the overly entitled and pampered boris and jrm class might think and the nonsense peddled by the Mail and the like (generally staffed by graduates from tuat class, when they're not buying in cheap stories from overseas press agency mills). Even the OU would have been out of my reach. I have several friends who obtained their degrees while working, but they had to wait until universities finally saw sense about part time degrees (very common in Germany even when I was in my 20s).
When I went to university I felt as if my poor background really showed - I'd never been abroad, let alone skiing or to America; and unfortunately some of the posher "gels" in the hall of residence really made me feel that, including the ones who were supposed to be the welcome committee on my first day - I arrived late and not in a car, so evidently not well off, and they were horrible to me. The warden had gone back to their house in the grounds by then and evidently these delightful madams didn't need to show how lovely they were any more...so many other small examples, not knowing about food, or what cutlery to use at our formal meals, or about artists or celebrities of the day, or anything about ponies or lacrosse, or what the brand of scarf you wore told the world about your social class (this was long before designer outlets and ebay!) I had made the mistake of choosing a very green welly university...so called as lots of the students were from very posh backgrounds where Hunter wellies and Barbour jackets were the order of the day.
And it definitely held me back - I didn't have the confidence to join the drama group or the student newspaper. I didn't even think I was allowed to use the computing centre until I realised it was for everyone in my final year. And when I got into difficulties in my 3rd year it didn't even occur to me I could go and ask for help from my tutors or the careers service.
It wasn't until I'd worked and mixed with people from outside the UK, especially Americans, Canadians, Australians and New Zealand (it was in an ESL setting) that the chips on my shoulders started receding, they didn't give a monkey's how many ponies you had, they really were more interested in personal merit and my ability to get on with others.
Now, years later when I've travelled and worked overseas and met people from all over who don't know my background and if they did wouldn't care, I look back and laugh at some of it. But that sense of inferiority still lingers quietly on, and I'm determined I'll help my younger sister's kids have that background knowledge they need to apply to higher education and thrive if it's the right option for them, or to be successful in other ways if not. And I'll also make sure they have all the information they need about grants and scholarships and essay prizes and positive discrimination due to postcode they might need to help them get on. It's not a coincidence that these grants often go to middle class kids and up - there's a lot of sharp parental elbows out there in my experience!
There are some totally free or very low cost tuition degree courses out there in parts of Europe, especially Scandinavia, and taught in English too, but those opportunities are no longer open now we're a 3rd country and not part of the EU...
Most people who move in what used to be called the middle class and up will never know that sense of inferiority, even if they don't consciously understand their level of privilege. I remember reading something a while back that pointed out that if Prince William was going to marry a commoner it would always have been a Kate Middleton type - an ordinary girl from the comp would have been far too bolshy and never have mixed in those circles anyway. Whereas she went to the right school, was intelligent but not TOO intelligent, well groomed and quietly confident, and also sufficiently sporty to be very acceptable in that circle of society.
There's a student from Bristol University who is from an ordinary working class background and who has founded a movement focusing on ex comprehensive students at university, sounds as if her experience of academia has been pretty similar to how I experienced it.
In terms of advice on how you can help your kids, reading widely including newspapers and not being put off by long words or political bias (read the Guardian AND the Daily Mail, the Spectator AND the New Statesman as as that gives a pretty broad spectrum, plus magazines- the local library should have an app called Libby or similar which us free and has magazines on all kinds of topics, and these days there are fantastic documentaries on YouTube and other places; and also building their confidence whether that's via sports or cadets or drama or public speaking of some kind. Places where they can mix with as broad a spectrum of people as possible will be a good start - my sister's kids went to scouts for a few years and it definitely shows in their confidence even now that they learned how to mix well (along with potentially useful post apocalyptic skills like how to skin a rabbit and build a fire :))
Despite the lip service about levelling up, social mobility in the UK is currently at an all time low and it's an absolute disgrace, but that doesn't mean your kids are any less capable or deserving of the chance to progress.
Sorry this is such a long post but I have felt so strongly about this my whole life and you only really understand it if you've been there.
All the very best to your kids and you, you sound like a super, lovely parent.