Yes, it is possible for this army to be protected from radiation.
The problem is, everything depends on the amount, megatonnage,location, and detonation height the bombs, not to mention how long its been since the bombs were dropped. For the sake of this question, we'll assume that we're talking about a total nuclear war taking place, since we are evidently talking about a full-on wasteland situation.
Now, at this point, we have to define what you mean by 'wasteland' If you're talking about a barren, Fallout/Mad-Max desert, then the conflict really could have taken place anytime from 5 to 20 years ago. During this period after a nuclear war, for the first 1-5 years you would experience a harsh and likely devastating nuclear winter, where the temperature and amount of sunlight would drop, which, when combining with the vast clouds of radioactive dust sweeping the country, would result in pretty much all plant and animal life dying, at least in North America and whatever other geographical locations were heavily nuked.
After this, it is possible that with the build up of greenhouse gasses and destruction of the ozone layer caused by the nuclear war, the world could enter a Mad-Max type nuclear summer, in which the vast plains are transformed to deserts, creating an even harsher wasteland. Huge radioactive dust storms, reminiscent of the dust bowl would sweep across these vast new deserts, carrying possible lethal doses of radiation with them. However, after 20-30 years, even if the world has not regrown its flora and fauna, by this point most of the radiation will have settled and dispersed to non lethal levels.
So, at this point, you have a couple of options:
Wait 30 years, and you've got no problem with radiation
If you want to set up camp prior to this 30 year period, everything comes down to geography. It will be necessary to set up camp in an area that is fairly distant from any contaminated site, and also UPWIND from the location. This will generally be west of the blast area.
It will also be wise to set up at a higher altitude, like at the top of a mountain range, which is unlikely to be as severely effected by radioactive fallout, as the radiation will settle mostly on plains and plateaus. Mountaintops will likely be free of severe radiation within 5-10 years.
You can also take advantage of the rain shadow effect by setting up camp in the rain shadow of a mountain range, which will prevent radiation from being carried from irradiated areas in the EAST into your camp.