OP, don't worry, universities are a minefield to navigate and I think it's great you are asking on here and getting more information. Don't go off a family member or friend's opinion unless they work at a uni or know that subject area or best, a combination of both. Look at the high-scoring unis in the league table for those subjects, and go to a couple of open days with your daughter.
There are no unis that you should take off the table because you perceive differences between state/private pupils, private school pupils are over-represented in the uni system for a whole complex of reasons and if anything receive higher offers. As someone else has said, your dd may qualify as a widening participation student if you or your husband did not go to university, it's not just done off postcode/deprivation indices. Each has its own criteria, so you need to check for offers, my dd received two out of five offers slightly lower due to this.
Also, you mention you have ASD, I don't know if your dd has ASD or is neurodivergent, but many students are and this is again something you might like to consider- is there good support for students on the spectrum? There have been threads on this in the past. Universities have to provide support, but some are better at this than others, and it's good to hear from other parents and pupils.
Good luck with it all, your dd sounds like a great student and I think with those grades she'll be able to go somewhere good, and you are a supportive mum which will also help her enormously.