The British were worried about a German dirty-bomb attack on London- hence the raids in November 1943 (USAAF) and February 1944 (Norwegian saboteurs) on the Norsk Hydro plant at Vemork. A few American officers were trained to use Geiger counters in anticipation of a German dirty-bomb attack on the D-day landings, but the Americans took the threat less seriously- and probably the same sort of way as gas. Arthur Compton's report to Vannevar Bush might be similar- it talks about "production of violently radioactive materials... to be scattered as bombs over enemy territory" as the easiest military application of nuclear fission, as well as mentioning power for ships and atomic bombs as applications that would take longer.The idea was most certainly not there at the time. More accurately, our current day appreciation for the hazards of radiation exposure were not even guessed at in the 1940s. The situation surrounding The Demon Core illustrates this neatly. Seeing as physicists at Los Alamos handled a critical mass of plutonium in such a cavalier manner, radiation hazards were obviously thought of very differently.
The hazards of very long term exposure to relatively high levels of radiation were known thanks to the fate of the Curies, radium workers, and others, but the hazards of sort and medium term exposure to what were thought of relatively lows levels of radiation were not appreciated until the mid-1960s.
The dirty bomb suggestions by both sides in WW2 were made with a chemical or gas mindset; i.e. the radiation dispersed works more by imposing a hindrance than by killing.
Bill
As far as fallout goes, the Frisch-Peierls reports to the Tizard Committee say that a by-product of an atomic bomb would be radiation "fatal to living beings even a long time after the explosion" and that "owing to the spreading of radioactive substaces with the wind, the bomb could probably not be used without killing large numbers of civilians".
This suggests to me that the British were worried about long-term radiological contamination.