The Soviets suffered enormously at the hands of the Nazis, and had every reason to hate them. It would seem that the Soviets should have been quite interested in finding and bringing the people responsible for this to justice. Not only that, but I'd imagine the propaganda value of finding Nazis being protected by the west would be immense.
I mean, if the Russians could say "Hey, look, Jewish people, NATO is protecting the butchers of our people and we are bringing them to justice" I'd imagine they could have one hell of a propaganda boon. It would be an excellent way to bring more Jews and Israelis in as sympathizers.
The KGB would have had significantly more resources than the Mossad, and the Soviet Union was a far more influential nation, so I'd imagine they could have done a pretty good job.
Yet, it seems they had no interest in it, outside of Stepan Bandera. Was there some sort of policy against it or did they just never bother?
Quite simply, the KGB had other priorities — even during the Second World War, Germany was never the Soviet intelligence services' primary concern. In fact, their greatest intelligence successes during the war were against their ostensible allies: Britain and the US. In KGB parlance, the US was referred to as 'the Main Adversary'; by the outbreak of war, it was clear to Moscow that the global balance of power had shifted, and that the US was a superpower-in-waiting.
Soviet mistrust of their allies was virtually on par with their contempt for Nazi Germany, to such an extent that it crippled most attempts at intelligence-sharing. In the most egregious example, the Soviet political and intelligence establishments ignored allied intelligence warning of Hitler's plans to invade the USSR in 1941.
For the Mossad (and for Israelis more generally), the campaign to bring Nazi war criminals to justice was a crusade: it was a very personal, emotional vendetta against the people who had attempted to annihilate the Jewish race. Dan Raviv's excellent history of the Israeli intelligence services, Spies Against Armageddon, covers this mindset in some detail.
The KGB's modus operandi was far more pragmatic; they were driven by ideology, not emotion — and fascism/Nazism as ideologies of any particular note were casualties of the Second World War. The new Cold War was a battle against liberal capitalist democracy, not fascism. By far the best histories of Soviet intelligence are Christopher Andrew's The Sword and the Shield and The World Was Going Our Way — I'd recommend checking those out if you're interested in how the KGB (and its predecessor agencies) worked.
They didn't hunt the nazis. The officially demanded extradition. Sometimes it was granted but sometimes, if the Americans have deemed the nationalist thinker somehow useful, they would claim that it is slander, and grant him protection.