You pretty much nail it, there were an unconfirmed report of Wagner approaching a warehouse that holds tactical nukes (I really hope that remains a simple speculation.) If Wagner claimed the tactical nukes, that would've very nearly guaranteed a successful coup, because they could simply waltz right into Moscow and hold tactical nukes to keep the rest of Russian army at bay and other factions in Russia would realize the same thing, "Oh wait, so if they can do it... then...."

Cue the Western Governments poker face at the enactment of the movie, The Dark Knight Rises, where terrorists/fascists gain access to nukes and Russia balkanize by competing warlords holding at each others throat with tactical nukes.