My sister is a narcissist with mood swings and bullies people. What mental illnesses could this be (she is aged 27)?

"Narcissism" IS a mental illness, or at least a personality disorder.

The mood swings and bullying, though, suggest something else.  We are not psychiatrists here, and even if we were, we could not possibly diagnose someone we've never met.  You've provided very little information in any case.

You can't force someone into treatment. All you can do is take care of yourself.  Focus on doing that and forget about trying to diagnose your sister.

Anyone that expects strangers on YA! to provide medical or psychological advice needs more help than their sister.