I'm not biologist but I will give it a try.
The songs of these birds can sometimes be used by other animals, because it can point a danger out (birds sing to mate and to communicate, sometimes when facing a danger like a predator). But this is not the major role of these birds, and i really don't know if their absence would change anything on this point.
The biggest role of these birds is determined by
* What they eat : insects and seeds.
There are many species eating insects (other insects like dragonfly, spiders, reptiles...), and without songbirds, they would have more to eat and would prospere a little more. If not sufficient, there might be other insects-eating species. Carnivorous plants ? Other reptiles, mammals ? Pick what you want.
Fruits and berry-eating birds have another role : scatter the seeds. There must be species that eat fruits and seeds and then go away. But there are already other species doing that (rodents, and even bigger animals, like humans are eating seeds and berries...)
Except a few more spiders and rodents, I don't think there would be a big change here.
* What they are eaten by
It's a bigger issue as the predators did evolve exactly to chase songbirds. Some birds of prey (like accipiters) would not exist at all. 445
There would probably be other predators species chasing the reptiles/insects/spiders... that took the place of song birds in the food chain. If they don't need to chase something that can fly, they don't need to fly. On the other hand, cats can chase songbirds when they find some, but if they don't they have mouses and rodents instead.
Some of the songbirds migrate, but I don't know wether it could change anything. But there are another birds migrating, not songbirds, then if they have a role in spreading plants/insects by changing continents, the job would still be done.
Conclusion : Without songbirds in the evolution, other species, known or not in our worls, would have taken their place. Nature keeps a good balance and would have evolved in another way. In fact (I would like a confirmation) for what I know and what I can find on the net, even if songbirds should disapear just know, nature could replace them with another species (it will take time), but it would kill their predators (mostly birds of prey as terrestrial predators could replace them with other small animals such as lizard or rodents). 445