We have to look at "Spring's" setting from at least two perspectives (more on a possible third later): the play it is part of and the poem-song itself. In the play, Love's Labour's Lost, the setting for this song is outdoors in the park or gardens of a castle, and we get the sense that it's a nice sunny day. So "Spring's" setting in the play is super-fitting for the song's content, all spring-y and flowery and colorful.
The poem itself definitely has an outdoors-y vibe. We feel like the speaker is out there in nature, reporting what he sees and hears. With all that discussion of flowers and birds, all that spring-y color and sound, it's easy to imagine a garden setting.
Now, about that third setting. Remember, "Spring" is sung as part of a play within Love's Labour's Lost. (See the "In a Nutshell" section if this is news to you.) In what setting did these players intend "Spring" to be sung? Well, the play they came up with was a kind of review with lots of different historical and mythological figures playing their parts. So the song "Spring" would have been sung after hearing speeches and watching scenes that include a mishmash of real and imaginary characters. Someone watching this play would have been sitting there with no idea of who or what he might see or hear next. This setting holds all sorts of surprises, kind of like the way springtime might hold a surprise or two for some of those "married men." (Cuckoo, cuckoo.)
In this way, the multiple perspectives from which we can consider the poem's setting reflect the multiple perspectives that the poem-song explores. Spring is a time of love and rebirth, but it's also a time of worry and suspicion. The cuckoo's song serves as a symbol of spring and all things spring-y, but it's also a mocking warning and reminder of the ugliness of infidelity.
Gee whiz—and we used to like springtime. Now, it seems super-complicated and we feel conflicted. Thanks for nothing, Will.