For many Romantic poets, including Wordsworth, the nightingale was a natural poet, the voice of nature. The nightingale is so named because it is one of the few birds that sings at night. In urban settings, the nightingale will sing louder to overcome the background noise. Being one of the few birds which sings at night, the nightingale has been used to symbolize individuality and a close, solitary connection with nature. Singing alone at night directly parallels the "solitary" reaper. The speaker in "The Solitary Reaper" notes that the reaper's song fills the vale much like a bird's song would. Therefore, there is a strong connection between the reaper and nature, just as there is a strong connection between the bird and nature.
The nightingale is also known for a wide range of notes, sometimes involving a crescendo flourish. This is a reflection of the range and drama of the reaper's song.
Cuckoos are also primarily solitary birds. Cuckoos are spread all over the world, except for Antarctica and drier regions. When the speaker in the poem realizes he does not understand the girl's Scottish dialect, he imagines her song coming across the seas from the Hebrides (in Scotland). The cuckoo is so widespread and in this sense, the cuckoo transcends space; it is everywhere. Then the speaker imagines that her song is also about things "far-off" which could imply that the song is about things far away or, in time, long ago.
Shelley and Wordsworth (and Keats) used birds in their poetry for these reasons but also because they saw a connection between the sound of bird songs and the rhythm and meter of poetry.