For me it's pretty simple: you have the opportunity to sign a player like Trent Alexander Arnold, you just do it, who cares about the manager? He's a world class player, he's still relatively young, he's coming for "free", you can't miss this opportunity. I'm pretty sure that any manager would love a player like him, there really is no reason to delay a deal like this.
It shouldn't be a problem when a manager leaves and every player who comes in should have the chance to adapt to the new manager. Trent Alexander Arnold is one of the players who is developing now and he has good skills so I don't think it's difficult for him to adapt to the manager and the style of play that the new Real Madrid manager will set. Any manager would love Trent Alexander Arnold's style of play and I don't think it's a big problem.