My understanding of the usual expression of cuckoos is that they hatch alongside the genuine nestlings and are then given all of the food, leaving the genuine ones to starve.
If that's right, abandoning the whole nest seems like a waste - it should still have viable wren chicks in it?
Particularly since the mechanism you describe involves recognizing the cuckoo as an individual chick (assuming there are also wren chicks in the same nest, they will repeat the song, so it's necessary to keep track of which chicks have done so and which haven't), it seems like a better approach would be to just kill it. Do you know if any wrens do that or might be moving in that direction?
(On second thought, I see that the National Geographic piece appears to say that the cuckoo chick will kill the wren chicks? That's not quite clear, since it also says the cuckoo will hatch first, so I'd expect it to be shoving "eggs" out of the nest rather than "siblings".)
In any case, fairy wrens are very fast reproducers (raising 2-4 clutches per year), so I imagine they just start over with a new nest when this happens.
Only about 40% of fairy wrens abandon cuckoo chicks, which suggests that this may be a relatively new development in these birds' evolutionary arms race.