I think it depends pretty heavily on how you see the transition between the games happening, what you think your players will be most into, etc.
That caveat aside I would absolutely definitely 100% not have players pick characters before playing TQY, were I doing this. But the fact that this is so clear to me suggests that we might just have very different notions of how the two games would interact, how the fiction would pass from one to the other, etc.
For example I would strongly suggest that all of the PCs in the AW game are members of the first generation of the community to be born after the arrival of the Frost Shepherds, to discourage players from trying to insert their future characters into the fiction of TQY.
I feel pretty strongly that the Quiet Year works best when social groups and dynamics are emergent from the map and the situation; trying to impose specific personalities (or even classes of personality) from the AW playbooks prior to that process of discovery seems like a waste of the game.
I also want my play of TQY to allow me to move fluidly between different community perspectives; I don't want to be worrying about which of these voices that I am temporarily taking on might be 'my PC' later on, and I similarly don't want to be tempted to assume that other players are becoming similarly entrapped by this expectation of future protagonism. And this is ignoring simple logistic complications like one player accidentally killing off the person another player was thinking they wanted to play later on.
Even if the goal is a much tighter continuity, I don't see much upside to pre-selecting playbooks; I guess the main concern would be that someone ends up wanting to play a book that has no obvious place inside the community & situation that are developed in TQY, but that seems like a problem with tons of easy solutions, most of which are likely to add interesting twists and introduce new dynamics to the situation -- which is what you're going to want to be doing anyways, as you move from one game to the other.
But just generally my goal would be to keep the doors as open as possible for unexpected transitions between the two worlds. Maybe after the TQY plays out the players decide they want to _play_ the Frost Shepherds, maybe the way things go down in TQY it's obvious that Brainers have some really specific social niche, or that it doesn't make sense for there to be a Driver, or whatever. Shutting down those possibilities before playing out the first of the games seems to undermine the potential of the experiment.