First off, that is a beautiful poem. The notion of the "zephyr" or "west wind" is common in more romantic styles of European poetry, and usually denote the soft, gentle breeze of summer or spring winds. I think the use of the word typically adds more of a mood than acts as a direct symbol, but it can function as a metaphor for many things. For example: the fecundity of youth and springtime, the whimsy of love, and the brevity and precarity of happiness. Sort of like how the rose symbolizes how beauty is ephemeral and mortal, the west wind captures the notion that many positive human experiences (such as happiness and love) are touching yet fleeting, like a warm summer breeze. For instance, if we look at the relationship between lying in bed with one's love and a west wind carrying rain, we get the sense of sorrow in the speaker, and the sadness of how we don't have the luxury of constantly being with our beloved; at some point, we all have to get out of bed.
Considering this poem is from the 16th century, as in, it's from the Renaissance era, it's probably at least in some part referring to Greek God Zephyrus, who had been written about in Greek poetry in antiquity. In Greek mythology, Zephyrus was the god of the west wind, the messenger of spring. There is some latent romantic notions in that Zephyrus is famous for attending to Cupid (Eros, god of love) in pursuing Psyche, so he is a literal attendant of love in this sense, on top of being the carrier of warm, gentle breezes. There are many Renaissance paintings of Zephyrus as well as poems because of his association with romance and the preoccupation of Renaissance artists and writers with Greco-Roman culture in antiquity.
Thanks for the good question!