I don't mean to be glib, but it sounds like there was some disconnect there.
Your sister didn't believe you, and she should have. She's also mad that you put him in time out. Did you have that sort of agreement with her, that you could discipline or punish her kids? I am going to guess that some people will feel that it goes without saying-- if it's a family member, an adult can discipline the child. I don't think it's a safe assumption to make, though. I have always chosen to either suggest to the child what MY limits were or to let the parent know about the behavior, but I wouldn't go so far as to put one of my nieces or nephews in time out. That wouldn't sit well with ANY of my sisters.
As for the lying-- that's really going to be her parenting issue to sort out with herself, believing her kid over other responsible, credible adults.
What's your relationship with this sister usually like? If it's usually amiable, try to connect with her in a few days after you have both calmed down about this. You both feel hurt. Personally, if my son is with an adult who needed to put him in time out for a safety reason, I wouldn't be as upset with the adult as I would have been with him, however, if there had been teasing and mean words, I would have preferred that the adult guide him and problem solve with him ("Suzy, can you tell Hulk how you felt when he said "XYZ" ) or gave him a boundary "if you can't be kind with your words, you may go play somewhere else. We don't speak to each other that way". The time out wasn't instructive in this case and it didn't solve the problem of him calling people names, it seemed like it was more of a punishment than a natural consequence, which apologizing or hearing the feelings of others would have been.
I hope you can work it out with your sister. Her son doesn't sound kind, I just think your sister needs to be approached in a way which doesn't raise her 'Mama Bear' defenses about her son.