I will say that whatever anti-christian ideas that Tukhachevsky might have, he won’t be openly expressing those as he will be using the Church as a pillar of power and Legitimacy. And no matter how much the Orthodox and Catholic Churches are different, the Orthodox priests would not remain silent if the Vozhd decided to purge the entire Catholic Church from Poland, they wouldn’t let a precedent for a possible purge against them be established.Will his anti-clerical tendencies lead him to purge the Roman Catholic Church in Poland wholesale?
I wonder how will the reverse Barbarossa play out. Can the Russian-Nazis use the same degree of harshness when dealing with the Germans that the Germans used OTL when dealing with the Soviets? The Russian-Nazis don't have the "crusade against communism" ideology to propel their genocide, and Germany is probably going to be much more liberal and democratic than the USSR was. If anything, Russia is going to be much more vilified than Nazi Germany was(due to the absence of a Soviet Union counterpart fighting against this Russia), or maybe they are just going to be less brutal.
While Tukhachevsky believes in Russian superiority, he is still Pan-Slavic. If I could exemplify by a hierarchy, first comes the Russians and second is their little Slavic brothers, several scales down there are the Germans and the Jews as second worse and worst respectively. And they can easily call a Pan-Slavic crusade against the Teutonic enemy, which would probably be even more convincing than a Anti-communist crusade of the Nazis IRL. And believe me, you wouldn’t like to be a German under the Vozhd, you probably would have better chances as a Slav in Nazi Germany.