I need some help? (Preferably from a parent or an older sibling) What would you do in a situation like this?!?

[Helen]

A lot of abusers 'pretend' to be nice. It's how they get away with it. Even if he's never been to jail, you can't say for a fact that he definitely isn't abusing her (even if it seems unlikely).
Either way, a very serious claim has been made and it needs to be addressed.
If your sister IS telling the truth, then she needs help, and your mum's boyfriend needs to be reported.
If she's lying (which I agree is probably the most likely scenario, given that she lied about him being in prison), then she needs to be disciplined very strongly about how dangerous such lies can be, and what damage they can do.
It's understandable you don't want to lose your sister's trust, but if her account is public, then you're covered. Anyone can read information that is posted in a public domain, and with so many followerers, it's likely that somoene who knows her or your family may read it. Your mum stands to find out anyway from someone else, so you NEED to tell her.
You can always ask your mum to keep quiet about who she found out from.
There's also the other option of talking to your sister yourself. Let her know that you read what she wrote and that she can't go around making up false claims as it can get people into trouble. Ask why she did it, and ask her to remove it and own up on Twitter that it was made up. You keep her trust, your mum doesn't have to find out, and everyone's happy. But if you do this, you need to be very clear with her and say all the right things. Tell her you don't want to lose her trust so you're giving her the opportunity to do the right thing, because what she wrote was serious, and if she doesn't take it down, you're left with no choice but to tell your mum. If you don't think you can handle it yourself or explain in the right way, then you are better off telling your mum and asking her not to say that it was you.
Those are your two options, but either way, you must do something before this gets even more out of hand and reputations get ruined.