TL;DR: It is meant by author to be set in indeterminable "close future to the reader" time, but can be pegged from textual clues to possibly be set in 1945-1957
As per Objectivism Reference Center's Atlas Shrugged FAQ
4.2 When is the story set in time?
The time setting for Atlas Shrugged is not specified in the novel. It is typically interpreted as being either the near future, or the present in an alternate version of reality. The social mores presented in the story suggest the US in the 1940s and 50s (when Rand was working on the novel), while the economic and political conditions resemble a cross between the early 1900s and a dystopian future.
Rand's own description of the time setting to one fan was as follows:
You ask whether Atlas Shrugged represents the present or the future. The answer is: both. To be exact, the action of Atlas Shrugged takes place in the near future, about ten years from the time when one reads the book. (Letters of Ayn Rand, p. 613)
Rand made this comment in 1963, and indicates a shifting timeframe: the novel is always considered to be in the future, regardless of when it is read, and therefore it has no specific time setting other than "near future." Reinforcing this idea of a non-specific time setting, Rand comments on the time setting in The Art of Fiction, saying that "Atlas Shrugged is of no period" (p. 163).
Both the futuristic and alternative history interpretations are discussed by Mimi Reisel Gladstein in her book about the novel, Atlas Shrugged: Manifesto of the Mind (pp. 40-44), and in Robert Hunt's "Science Fiction for the Age of Inflation: Reading Atlas Shrugged in the 1980s," in the anthology Coordinates: Placing Science Fiction and Fantasy (pp. 85-87).
The FAQ is not official, the ORC is not affiliated with Ayn Rand Institute or any other organization.
However, Conservapedia's Atlas Shrugged Chronology article took the pains to assemble a set of textual clues that place the novel in the years around 1950s.
I won't copy every bullet point of the article (which is worth reading for its painstaking detail), but the reference things such as state of transportation technology (rail vs. air service), telecommunications (phone access vs telegaph, radio vs. TV), cyclotrons, etc...
The most likely span of the years of the great strike is 1945-1957, with the novel's major action taking place in the last three years of this period. This novel, then, qualifies as an alternate history of the United States of America, and of the civilized world. In this version of history, the United States did not intervene in the War of the Reds and the Whites in Russia, and probably did not participate in the First World War. That War probably ended with every country on earth, except America, becoming a Communist state. In addition, prohibition cannot have occurred, because alcoholic drinks flow freely in many settings.
Politically, the United States would have been in the hands of the proponents of Progressivism, which was in fact a potent-enough force in the election of 1912 to field a third party candidate for President (Theodore Roosevelt) who actually out-polled the losing major-party candidate in that race, William Howard Taft.
(they speculate that Mr. Thompson was the anti-Harry-S-Truman based on Rand's own statements).