Who has family in law enforcement? Your GF? That shouldn’t matter. What she did was against the law. Call the police. Mom may need meds and special care that your sister can’t provide. She’s putting Mom in danger.
I’m not certain, but if your mom is not competent to make her own decisions, what your sister did was kidnap her. I would ask at the NH who witnessed your sister removing Mom from the home. Even if she went “willingly”, your sister probably had to bribe or otherwise coerce your mom into leaving. She may even have told the staff she had your permission.
I’d call the police.