The girls in school are singing a song that Stanley Holloway later sings in "The Titfield Thunderbolt." Does anyone recognize it?
T.E.B. Clarke presumably chose the same song for both his screenplays, unless Holloway was so taken by the song that he picked it himself. It was "Jolly boating Weather" and is usually sung by Eton School.
How are the Lavender Hill Mob caught?
We don't know, only that they are. It is usual in Ealing Comedies that the criminals get caught.