I remember reading this poem when I was a kid in the 80's, but it was likely older than that. We had to read it for school. I don't remember much about it, but the poem walks you through a suburban neighborhood where everybody is gone. It describes how the sun comes up, all the automated machinery comes on, with nobody to use them. A dying dog is wandering around in the neighborhood. It even described the shadows of schoolchildren burned into the brick wall they were playing in front of. I don't think that the story directly said, but of course back then we all knew that the idea was that a nuclear bomb had gone off nearby.
You're referring to the poem "There will come soft rains" by Sara Teasdale.
The poem was used in the Ray Bradbury story of the same name which went into greater detail (notably about the family dog and a "Hiroshima shadow" of a local family).
"There will come soft rains and the smell of the ground, And swallows circling with their shimmering sound;
And frogs in the pools, singing at night, And wild plum trees in tremulous white,
Robins will wear their feathery fire, Whistling their whims on a low fence-wire;
And not one will know of the war, not one Will care at last when it is done.
Not one would mind, neither bird nor tree, If mankind perished utterly;
And Spring herself, when she woke at dawn, Would scarcely know that we were gone.
I believe it is There Will Come Soft Rains , one of Ray Bradbury's works.
The poem is by Sara Teasdale but there is also a short story with the same title by Ray Bradbury in which he uses Sara's poem.