When should the second deck be shuffled if the card "Winter is coming" flipped during the Westeros phase? Should it be shuffled right after all the Westeros cards revealed or after the first card been played?

Resolve the card when it is that card's turn to be resolved, which would be after the card from deck 1 is resolved.

This does create the interesting scenario that you have already done the 'moving the wildling threat token' step, so, in theory, if the card you get after shuffling has a wildling symbol on it, then it is ignored.

Do other people play it this way, or does anyone move the wildling threat token again if the second card has the wildling symbol?

I agree that the new card should be revealed when you would resolve it (that is after resolving the deck 1 card), but I think the wildling threat is added.

While the rewording of the 2nd edition rules make it sound like you would skip it, the player aide makes it pretty clear that you should count the new wildling threat (as in 1st edition). Presumably that could also cause an immediate wildling attack if the threat went to 12. (That's not addressed in 1st edition, since there was no immediate wildling attack rule).

The rule book states that all 3 Westeros cards are first revealed, then the wildlings track is advanced based on the number of symbols on the three cards. If this action results in the token reaching the "12" spot on the track, then a wildling attack is immediately resolved. Finally, each Westeros card is resolved in numerical order. The step of advancing the wildling token has already taken place, so I would say that another symbol being revealed would have no effect on the position of the wildling threat token.