depends which USSR countries you are talking about. If you're talking about client states like the DDR and Poland or Czechoslovakia, Hungary, Romania, Bulgaria, Albania etc., they didn't really divy anything up. many soviet generals, military officers, leaders etc. left the soviet union to return to their home countries, others stayed in russia (it was a pretty diverse federation/union to begin with, and the borrowing of ethnic leaders was common).

With the stans, turkmenistan, tajikistan, uzbekistan, kazakhstan, kyrgyzstan, they broke up, formed independent states, and there was alot of shuffling of weapons and a little bit of chaos as former Soviet political bosses swept into power and established dictatorships. as this happened they consolidated the weapons and formed their own armies from tribal militias that supported them, or some republics had standing police/military forces that remained largely intact. afghanistan was abandoned prior to the collapse of the ussr, but many of the conscription soldiers who served there went back to their respective countries and helped form the core of their armies and militaries.

in other republics or autonomous regions, such as in the north caucasus - ingushetia, chechnya and the like - they did not gain independence, and after some separatist/civil wars, the rebellions were quelled, and mini-dictators were established to control them. these are ran much the same way today that they were throughout the 20th century, using soviet/russian military equipment and military advisors, usually supported or propped up by ethnic soldiers of their own countries. So kadyrov runs chechnya with chechen soldiers and Russian made AKs and armaments.

Most of the nuclear warheads and nuclear technology were bought up by Russia following Russia's reconsolidation into the Russian Federation, so while Russia is today a nuclear power, the stans are not.