(OTB tournament) What am I supposed to do when I checkmate my opponent and they just stare at the board?

So usually, when I checkmate, I wait for my opponent to offer a handshake (which usually happens immediately). I don't want to offer the handshake myself, I think it'd be rude.

However, yesterday, I played my high school's chess tournament (time control was 20+0) and in two different games the opponent just kept thinking after I checkmated them. The first one tried to make a move, not noticing it was illegal, and after I called it out they noticed it was checkmate. In the second one, I akwardly said quietly after ~30 seconds "it's checkmate", after which they offered the handshake.

What'd be the "ettiquete" here? Wait until they realize or make an illegal move or run out of time? Quietly tell them? Help.