If the Catholic Church takes an anti-Nazi lone it will be suppressed, period. If it continues to be pro-Nazi on balance, I can't see countries that are in a cold war/anti-Nazi allowing funding for the Catholic Church to go overseas - for local efforts/parishes/schools yes, but to go to Rome and support pro-Nazi "propaganda"...doubt it.
In Europe, most established churches receive/received state funding. If the Nazis reduce this, cut it off, then they need to go to their parishoners for funding. As the number of real believers diminishes, so will funding. Also expect seminaries to come under more and more direct Nazi control...funding, curriculum, etc. Remember the tenets of Nazi philosophy go against the grain of more modern Christianity, so with time as the population is indoctrinated in Nazi philosophy from an early age, the attraction of Christianity will decrease, and agnostic/atheist or even pagan will become more popular with the "negative" career effects of Christian belief.
In Europe, most established churches receive/received state funding. If the Nazis reduce this, cut it off, then they need to go to their parishoners for funding. As the number of real believers diminishes, so will funding. Also expect seminaries to come under more and more direct Nazi control...funding, curriculum, etc. Remember the tenets of Nazi philosophy go against the grain of more modern Christianity, so with time as the population is indoctrinated in Nazi philosophy from an early age, the attraction of Christianity will decrease, and agnostic/atheist or even pagan will become more popular with the "negative" career effects of Christian belief.