With sufficient time, I think I might have the flights for leg 1. Details under the spoiler-
For teams traveling CLT to LHR, there's only two possible flights:
AA 730 CLT LHR 1755 0700 (+1) or
AA 732 CLT LHR 2100 1000 (+1)
For teams coming in from ORD, one of the vids Kiwi posted has Michael & Moe in terminal 3 of Heathrow, which means they would have to be on a BA or AA flight as opposed to United which arrives in terminal 2. Given the timing of the sighting I can rule out the one daytime flight arriving in LHR the evening of 2/22, leaving one of these flights which arrive the next day:
BA 294 ORD LHR 1710 0650 (+1)
AA 86 ORD LHR 1850 0840 (+1)
BA 296 ORD LHR 2035 1015 (+1)
AA 46 ORD LHR 2220 1205 (+1)
For teams traveling from JFK, the
picture of Ray & Caro and the others is in terminal 7 (courtesy of
this vid at 0:48) which is the terminal for British Airways
planes - whether it's an actual BA flight or if it's an American codeshare is uncertain.
The one thing that's definite is that it's an overnight flight (Ray & Caro are socializing at night), leaving us with 9 flights arriving between 0620 and 1035, all arriving in terminal 3 to meet up with the other two flights. I have a feeling that they might have ended up on one of
BA 174 JFK LHR 1850 0650 (+1) or
AA 104 JFK LHR 2230 1030 (+1)
For LAX, there's not a lot to go on but there are two patterns that formed above - one in which flights arrive in LHR at around 0700 or 1000, and another in which all flights land in terminal 3 (which all happen to be either BA or AA flights). Assuming one of these holds the possible nonstop flights are
VS 008 LAX LHR 1500 0945 (+1), which fits the time but not the trend of everyone flying on American or OneWorld affiliated airlines
AA 108 LAX LHR 1825 1310 (+1), which fits the airline pattern but is substantially later than the other flights, plus
VS 024 LAX LHR 1920 1335 (+1), and
BA 268 LAX LHR 1935 1420 (+1)
(this assumes also that teams flew nonstop from LAX; if they connected through ORD or JFK then this can be ignored)
My best guess is one of these two 'sets' of flights used for the majority of teams depending on when they departed their hometown:
which gives roughly equal departure times for the flights to London and can partly explain the long production hold.