There is no really simple answer to this. On the one hand, no experience is universal. The poem is written in English, describes a pastoral British childhood, and has specific references to the scenery, wildlife and circumstances of Thomas' childhood. Even the reference to Adam, which seems to universalize the poem, limits it to a western, Christian context. Moreover, the concept of childhood as a period of freedom, play, and innocence is a modern, middle-class western one. In sub-Saharan Africa or the southern United States before the twentieth century, young children in poor families might spend the morning fetching water from the well, tending livestock, and working in the fields, and far from living a carefree life would be focused on the source of their next meal. Even now, a child living in the favelas of Brazil or slums of Mumbai does not have a life remotely resembling the one in the poem.
On the other hand, although the poet's imagery is localized in the Welsh countryside of his childhood, he does attempt towards the end of the poem to universalize the experience into a general one of loss of innocence. Although the childhood images are localized, and we can infer something about the child's class background, the child is simply referred to as "I" and no biographical details are offered. The pastoral imagery and mention of Adam universalize the theme in the sense of associating the youth and innocence of the child with Adam and Eve in the Garden of Eden and aging with the same sort of fall from innocence that happened when Adam and Eve ate of the fruit of the tree of knowledge. Thus the poem strives to convey some sense of universality in the portrayal of how time affects people.