If you listen to the songs on Spotify - the contestant versions, you'll find they are clean of applause or FX and all have clean outs.
So obviously those are recorded somewhere in advance if they appeared straight after last night show.
It's not unthinkable that a direct mic feed would cut out audience applause. It's far more likely that they were recorded in the dress rehearsal however.
EDIT: Confirmed by a spokesperson (as the tracks were uploaded this week before the performances aired):
"The songs are recorded earlier on during rehearsals as practically there isn't time to take the recordings from the show and put them on Spotify straightaway."