I've never heard of any kind of "drug cocktail" that would do what you are talking about. In general, if you were suffering from symptoms of radiation exposure, those would be treated in a medical facility. Depending on the severity of the exposure, those symptoms might be quite mild and easily treatable (like gastrointestinal distress, for which there are many medications to treat) or might be severe and life-threatening (and require hospitalization, or just render you totally incapacitated). If you are interested in the doctrine on this, the NATO Handbook on the Medical Aspects of NBC Defensive Operations from 1996 is probably essentially the same as what the 1980s procedures would be, and goes into great detail.

Games like Fallout aside, there is no magic bullet medicine that will make you recover from, or avoid, exposure to ionizing radiation. The closest one gets are things like potassium iodide, which are useful if you know you are going to be exposed to fresh fission products, because they will prevent your body from absorbing I-131 into your thyroid; that can help, if you know in advance that fresh fission products are coming your way, but I-131 is only one fission product of note and so this is not a "universal" solution. Note that the above-linked manual makes clear that NATO guidance did not include the stocking of such things.

The way to avoid ionizing radiation exposure is to not be around when the nuke goes off, or to have enough matter between you and the radiation source that most of it gets absorbed by it. If you ended up in a situation where you were exposed to a likely-fatal amount of radiation (say, 5 Sv/500 rad), you might still have some combat effectiveness in you for awhile — it takes VERY high exposures to render a soldier near-instantly incapacitated, much higher than fatal levels — but it is going to degrade over the next few hours and days and then you will be just another casualty of this war. This sort of thing was taken into account by planners on both sides for tactical nuclear warfare, as they intended to expose the other soldiers to high-enough levels that they would be rendered essentially incapacitated (this is what radiation-enhanced weapons — the so-called "neutron bomb" — were intended to do; they weren't trying to kill troops slowly, but very quickly).

Any exposure over 1 Sv/100 rad comes with likely radiation sickness, and an increased cancer rate if you survive it, along with a number of other common long-term effects like cataracts. Exposures under 1 Sv/100 rad have probably no acute, immediate effects, but do have a likely small increase in lifetime cancer risk. Again, that wouldn't be that relevant for your scenario — the Soviets don't care about your lifetime cancer risk, and let's be honest, if you are in a shooting war with the Soviets the odds of you dying of a long-term cancer are pretty low, because your life expectancy is probably going to be in the range of hours anyway at that point, from other factors.