Our speaker of "Who Has Seen the Wind?" sounds like she's telling a nursery rhyme because, well, she is. Maybe she's a bit more eloquent than the speaker of Humpty Dumpty, but since Rossetti was known more so for her countless lyrical poems and ballads than nursery rhymes, we can understand the extra poetic pizzazz. So although our speaker fits the mold of your typical nursery rhyme with a first-person voice, we can easily pick up on the little nuances that make this rhyme a bit more sophisticated than most.
After all, we don't often see the neither/nor clause in a lot of nursery rhymes. We're also not used to speakers subtly turning syntactical structures around to throw us for a loop, like we see in lines 2 and 6: "Neither I nor you" and "Neither you nor I." Usually speakers of nursery rhymes are a bit more obvious with their language play like the kind you see in "Incey Wincey Spider": "Climbed up the spout/Down came the rain" (source).
If we were to describe Rossetti's speaker of this rhyme, we'd say she's a refined nursery-rhymer with clever lyrical chops. She's not looking to be too cute in the way she sounds, hence the slightly more sophisticated clauses and hints of figurative language: "leaves hang trembling," for example. Maybe if we were to look for a Disney comparison, we'd say our speaker is like the Fantasia to your Dumbo (not that Dumbo isn't totally awesome).