Although the sentence seems wrong to me, it is trying to say that while Naomi is waking up, she hears the sound of birds. The birds are not causing her to wake up - they are just there.
The sentence seems wrong to me because the verb "awake" is not normally used this way. It is normally used as a transitive verb meaning "to cause someone to become awake". When I read "Naomi would waken..." it makes me wonder who it is that Naomi is waking up.