Three year old throws a tantrum after being forced to apologize. What to do?
Sounds like your average 3 year old to me. The best way to keep a lid on toddler tantrums is by preventing them from happening in the first place (as much as one reasonably can). That means not dragging your child out on errands when they're tired, hungry, or sick, or forcing your child to participate in social situations that make them uncomfortable. It also means utilizing a lot of distraction and redirection, as well as being in tune with your child's emotional state and recognizing his or her emotions as valid. Your nephew is a person, not a trained poodle, and while the emotions he's experiencing may seem over-the-top to you, they are very real to him. Also, while it may seem like your sister's attempts at using time outs "don't work," it's actually working fine. She is isolating him for a few minutes so he can calm down, OR telling him to go to his room if he's going to keep pitching a fit. In either case he's being given the opportunity to practice regaining emotional control, while communicating the message that if he's going to behave poorly around others, he will be made to leave whatever activity is going on. And after a little break, he can come back and try again. Sincere apologies aren't going to happen at his age since he's only 3, but he's still being taught the convention that if you offend someone, you apologize. Sincerity will come later when his sense of empathy is more developed.
He's 3, so good luck. I don't think it's terribly important that he apologize in the moment as long as he does so afterward. A 3 y/o doesn't have the capacity to consider anyone other than himself when he's upset. The time outs tell him that the tantrums don't work and that's probably the best you can do.
There are books out there called Child Training Tips. But a three-year-old typically acts that way. he is learning self-control and the consequences of not practicing it. Give him time. You might take him aside and tell him that when he acts good, he will be rewarded, and when he acts bad, he will be punished. At that age, even a humiliating spanking might teach him better than just sending him to his room, where he can throw more tantrums and let his selfish pride rage unchecked.
At least he apologized....unlike Trump.