In my opinion, this poem is about the poet's attitudes towards death. He rejects the way that people usually think about death. To him, death is just a natural process, one in which the person goes back and becomes one with nature.
You can see this, for example, in the second stanza. There, the poet is saying that he will not really acknowledge the child's death until he, too, is dead. But look at how he characterizes death there. He says that, in death, he will "enter again" the round bead of water and the ear of corn. He is saying that he has come from natural things (water and grain) and will be going back there.
In other words, death is a return to nature and he will not presume to mourn the child's death because it is just part of a natural cycle. Through the rest of the poem, the poet restates this theme in various ways.
Despite the fact that he lived during the period of Modernism, Dylan Thomas is considered a Romantic poet. Indeed, his poem "A Refusal to Mourn the Death, by Fire, of a Child in London" is quite similar to another poem entitled "Thanatopsis" by the Romantic poet William Cullen Bryant (1794-1878). "Thanatopsis" suggests that one can see death as a return to nature. In the poem, nature reclaims the dead, who lie together in the earth.
With patriarchs of the infant world—with kings.
The powerful of the earth—the wise, the good,...
All in one mighty sepulcher....
Similarly, Dylan Thomas perceives the girl who died in a fire, "London's daughter," as lying with "long friends." In other words, she lies along with the others who have died before her. The dead girl lies with the "grains beyond age," a symbol of time, and the "dark veins of her mother," a reference to the earth and the rivers ("veins") that sustain it.
Clearly, Dylan Thomas believes that nature has regenerative powers, and this idea is expressed in his poem. It is because of this belief that in the stanzas 1–3, the speaker declares that he will not mourn the child's death until the end of the world comes—"the still hour / Is come of the sea tumbling in harness"—and he himself dies. Unlike what was done at the girl's funeral, the speaker refuses to eulogize her or speak tragically of innocence and youth.
In the last stanza, Thomas expresses the theme to which the ideas in the previous verses have been leading: as a person's body becomes part of nature, it acquires regenerative powers. The young girl lies "Robed in the long friends" of earth and water. After her "first death," Thomas contends, "there is no other" because she will become part of the earth and rivers—matter that can never be destroyed.