In The Nuclear Express by Thomas C. Reed and Danny B. Stillman, they explain that the term "nuclear fission" was first used in Physics by Lisa Meitner in a letter to Otto Hahn on December 19, 1938. How quickly did the term spread? How did this term spread?
From the book, Before the Fallout: From Marie Curie to Hiroshima there is this comment:
Bohr urged Frisch to write a paper with Lise Meitner as soon as possible and promised to say nothing until it was published. By dint of long-distance telephone calls, aunt and nephew drafted a short note to the editor of the British journal Nature, describing the splitting of a nucleus and the theory underlying it. They also found a name for their new phenomenon. Frisch asked an American biologist, William A. Arnold, working in Bohr's institute, what he called the process by which single cells divide into two. He replied, "Fission".
(this was in Dec. 1938).
Certainly by the end of 1939 the term was used a lot. Some papers:
The Mechanism of Nuclear Fission. Niels Bohr and John Archibald Wheeler. Phys. Rev. 56, 426, 1 Sept 1939.
Resonance in uranium and thorium disintegrations and the phenomenon
of nuclear fission. Niels Bohr, Phys. Rev. 55, 418, 15 Feb. 1939 (!)
The Fission of Uranium. H. L. Anderson, E. T. Booth, J. R. Dunning, E. Fermi, G. N. Glasoe, and F. G. Slack. Phys. Rev. 55, 511, 1 Mar. 1939
Number of Neutrons Liberated in the Nuclear Fission of Uranium. H.
Von Halban Jun., F. Joliot and L. Kowarski. Nature 143, p. 680, 22
Stability of Uranium and Thorium for Natural Fission. W. F. Libby,
Phys. Rev. 55, 1269, 15 June 1939.
Products of the Fission of the Uranium Nucleus. L. Meitner, O. R. Frisch, Nature 143, 471-472. 18 March 1939.
Actually, DOZENS of papers on nuclear fission came out in 1939. Just try Google Scholar, Fission and 1939. Probably immediately after Rosenfeld's talk at the Princeton physics club in mid Jan 1939 where he passed on word about nuclear fission (that he got from Bohr and Wheeler) the phrase spread throughout the physics community.