She does come back, eventually. She waits it out as long as she can before the freezing weather drives her back inside but even then, she returns in silence and seems intent on remaining that way for the rest of their time at that particular post. That's about how well the rest of the Long Night passes, too, though Daud will find himself with a much closer shadow than before while his emotions and animas are both still missing.

Billie doesn't say much during that time, either, if only to save herself the headache of asking questions and getting uncaring answers. It isn't as if Daud had ever been some fountain of caring thoughts and friendly conversation but even his normal level of stoic observation was miles better than this. It means she does her best to settle into a similar lack of emotion but she's by far worse at it than he is.

Why? Because unlike Daud, she does still have her animas and Deirdre frets and paces and preens nervously in her counterpart's place.