Finally! So I got the cutscene with Nozomi and rang Master Chan. I went through all the cutscenes, started a new day, spoke to Fuku-San about money to get to Hong Kong (you can't leave the house until you do this), and set off into Yamanose. It was snowing.
I saw Megumi staring at an empty box, spoke to her, and, hey presto! It triggers the sidequest.
This means two things:
1. You don't have to trigger the second cutscene about the kitten's leg as I only triggered the cutscene with Nozomi and that seemed to do the trick. That cutscene, like the kitten's leg cutscene, can only be triggered when it's not raining, snowing or night time.
2. It doesn't matter what the weather is like after you've taken the mirror to Master Chen. The missing kitten sidequest is not triggered automatically, but is triggered after a conversation with Megumi the day after you have spoken to Master Chen.
I really hope this helps somebody else because it's taken me hours!