In that days they have no idea of radiation poisoning , so they will keep using that air base.
They knew about acute radiation sickness, although they didn't have much in the way of details. And they knew radiation caused cancer, but they didn't know how much, and believed there was a threshold below which it was safe. (For that matter, we're still not certain there isn't...)
They'll know enough to keep people away until radiation has died down to levels that won't cause short-term symptoms. Take a look
here for a map of fallout plumes from a 10-kiloton groundburst. The purple zone is the area where radiation dosages exceed 10 R/hr. Note this does NOT include the dose from the initial pulse, just fallout. You need to absorb a cumulative dose of about 100 R to have any risk of suffering short-term effects. Just eyeballing it, it looks like most of the island will take a cumulative dose sufficient to cause acute radiation syndrome. That doesn't mean everyone will die, mind you, or even most people, but some will.
Long-term, radiation will decay as approximately the (-1.2)th power of time over the first two years. After a month, areas with a dose rate of 10 R/hr at two days after the blast will drop to a dose rate of 0.31 R/hr.
As far as cancer effects go, I happened to look up the US evacuation plans for Trinity for another project recently. The government planned to start evacuating towns if projected cumulative dose exceeded 75 R. The current EPA limit for civilian radiation exposure is about 0.1 R. According to the NAS, 75 R will ultimately cause cancer in approximately 7.5% of the people exposed.