The most important thing we need to know before answering this question
is the location of the players relative to population and industry
concentrations that will be hit, how this relates to typical wind
patterns, and what the local seasons are.

To pick on that last point:

* If you're in the summer in the hemisphere where the players are, there
will be more atmospheric moisture (in an absolute sense) that might be
triggered as rain.  You'll have one set of relatively immediate effects
of dust-seeded rain downwind of immediate strikes leading to possible
localized flooding, followed by an effect a day or so later a much
broader atmospheric dump as additional cloud cover (vapor from sea
strikes, smoke from many types of fires) cools the atmosphere.

* If you're in winter, much less immediate effect.  However, the dust
would be thrown on top of the current snow pack across the majority of
altitudes, which would lead to snow melt and avalanche as the snow packs
down due to this effect and maybe some seeded rain as above.  This would
raise the rivers over time -- and strikes might have already weakened
downstream dams.  The benefit here is that the first hours and days of
radioactive decay would be spread out across a larger area, and not
along the watercourses where population was probably gathered.

I think you'd probably find a search for Carl Sagan's description of
nuclear winter to be quite useful as well.

Alan

--
Bicycles are the most efficient machine yet invented for turning energy
into motion. Indeed, the bicycle has been accurately described as a kind
of green car, which can run on tap water and tea cakes and, moreover,
has a built-in gym.

Lord Taverne, UK House of Lords, in Nov 2015 debate