OP: This was me. My grandfather paid for my Dbro to go to boarding school (totally beyond my parents' means), I and my sister went to local schools. Perfectly good local schools, as it happens, and we all did fine, but I am the one with 2 postgraduate degrees, I am the 'academic' one in the family, and it still rankles a bit. We're talking 20 years down the line, here!
I'm not being petty, I don't think. I have a very good relationship with my parents and my siblings, very affectionate, very comfortable - but the fact is that my brother got preferential treatment because it was assumed (by my grandfather) that sons-and-heirs needed a proper schooling, whereas daughters didn't.
My mother was quite open about it when I asked her in my teens, she said she felt it was terribly unfair to us girls but that didn't justify her denying my brother the chance when it was offered to him. Now, had it been my parents who had made that decision, rather than taken up an offer from my grandfather - I think it would have affected our relationship in the long term, certainly. Your SIL is bonkers if she doesn't think she needs to explain herself to her daughter; she needs to if she wants to avoid hurting her daughter deeply, and risking a permanent alienation.
Tell your niece, btw, that if she's thinking Oxbridge then coming out of a state school can be to her advantage depending on which colleges she applies to. Many oxbridge colleges want to boost their state-school acceptance rate, and if she can get some sympathetic careers advisors to help her with mock interviews etc (or even better some oxbridge grad friend-of-friends) that will help her gain some of the lost ground that private school pupils will certainly already have been given.
It's certainly true that a big-name public school can help you in life but not all fee-paying schools have the same heft/clout. What they do give you is extras (music, drama, interview practice, teachers on-call after-hours to help with work, etc) which it sounds like your niece would have loved. And a safer environment for her to study, the poor thing.
If I were in the deeply unpleasant role of peacekeeper, I would suggest, gently, to your niece that this isn't a reward for her brother's slackness, it's remedial help. Tell her she is brilliant to have done what she's done, and she will always know that no one ever gave her any special favours, it was all her hard graft, and that counts for more (and may well count for more in uni applications).
Then tell her parents that she deserves a big, big slice of this suddenly-discovered money-for-education pie. Tell her that YOU will be (or make sure her parents are) on hand to make sure that things a private school pupil would have had on their CV (debate club, directing 6th form play, orchestra, work experience with barrister, captain of sports team) will also be on hers, because you will make sure her parents start coughing up for that stuff now. If they can afford school fees, they can afford grooming/coaching for their applying-to-uni daughter.
I'm fairly sure she doesn't want to deny these benefits to her brother, but it is monstrously unfair that they are being given to him and not her without any explanation. It seems like she is being left to sink or swim alone, while he gets a cushy life raft. So, while she doesn't want him to drown, why was there no help for her? Does no one care? That is the answer your SIL owes her daughter, otherwise she will make up her own answer, and it won't be one that is conducive to good family relations.