I literally cannot fathom how your mother abandoned your sister in such a way... and believe it. Did your mother go NC with your sister - or did your sister cut you all out of her life when she realised that not one of you believed her?
Others have already stated that many issues with MH are triggered either by childhood sexual abuse, or by the comprehension of it having happened. As others have said, it usually resurfaces when you become a parent, or an aunt/uncle yourself. And it doesn't happen to all of the children in a family. Your sister isn't mad, she isn't nasty, and I doubt very much that she infered your father had abused her just for the hell of it.
I believe your sister.
I suspect that she went NC with you all round about the time that she became a parent of small children, herself. That stage where you look at your toddler, and you think how perfect they are (even if they're really not!), and you know... you just know that you would do absolutely anything to keep them safe. And then you start wondering why your own parent(s) couldn't keep you safe - from one of them, or from someone else. And suddenly you're in counselling, and your own family are doing one of two things. They're either supportive of you, because they believe you - even if they don't understand - or they're adding to your pain by telling you that you're nasty, a malicious and manipulative liar, and insane, and trying to rip the family apart just because you feel like it... and you have no choice but to go NC with them for the sake of your own sanity.
My daughter has severe MH issues, and she is a survivor - it turns out - of childhood sexual abuse, carried out by her paternal grandparents. I'd suspected something had happened for a while... but my daughter was still in denial. Just insisted that she was choosing to have absolutely nothing to do with them ever again (her biological father hasn't been in her life at all) - which I respected. She had a mental breakdown about 18 months ago now, and is in counselling on a regular basis. During which the sexual assaults upon her came to the fore. She is very angry with me, because I "allowed" her to go to their house unsupervised from the time she was tiny. I didn't allow any such thing. She was left in the care of my parents, whilst I was working towards a better future for her... and they allowed them access. She has thrown accusation, after accusation at me. She's assaulted me. She's tried to take her own life repeatedly. And whilst she also has NPD, which tends towards making everything about her... not once have I told her that I don't believe her.
Because I do.
Even after everything which she's done to me - and believe me, it's been pretty bad over the last few years - not once have I abandoned her the way that your mother has abandoned your sister. I couldn't. I wouldn't. Because she is my daughter - and I would give my own life in a heartbeat if I thought it would take her pain, what happened to her, away.
I don't know if, in a few years time, my daughter will still be in contact with myself and her brother. She's already threatened NC several times when she's "free of [me] and [my] awful abuse" (basically, I expect her to pick up after herself... she is 21, after all!) and as much as that will hurt - I will respect her choice. If that happens, and I die, then I expect her brother to at least let her make the decision as to whether she wishes to grieve my death... or celebrate it.
Your mother is, in all probability, not being entirely honest with you, OP, and I would suspect that she's not being honest with herself, either. Whether your father abused your sister, or not... he didn't abuse you - and again, that's quite common. My mother was sexually abused as a teenager by her stepfather, yet her younger sister (his biological child) was not. It happens. And even though my grandmother stayed in her marriage for another 40 years... she believed my mother and my aunt. They continue to have a relationship to this day.
Shifting the burden of proof/responsibility onto your shoulders, and those of your other siblings, though... that's cruel. Let your mother continue to believe her version of the truth, by all means, OP... but at least offer an olive branch to your sister so that you can know the actual truth of her life.