Both the Italian fascists and the German Nazis were into Nietzsche a bit. Though I think they were mostly attracted to his superficial rhetoric extolling war and violence, rather than the technical elements. And of course, the Nazis embraced Heidegger, though I don't know enough about his philosophy to know if it was seriously fascist, or if he just added in some pro-Nazi stuff in order to prosper under the regime.
For the most part, I think there's a danger in overstating the importance of fascism's supposed philosophical underpinnings. Fascists in different countries took whatever inspiration they could from whatever sources, and looked at globally, these sources were often contradictory. The clericalism of a Franco(if we classify him as fascist) would be pretty uncomfortable(to say the least) with the Nietzscheanism of a Mussolini.
EDIT: For the OP scenario, maybe for some reason the Nazis promote Heidegger to chief-ideologue, and he proceeds to imbue their rhetoric with a lot of existential themes and imagery. But even then, I think it would just be a case of the Nazis using whatever-weapon-at-hand to build their rhetorical arsenal, without really caring too much about what it all meant.
An interesting book about the philosophical basis of fascism is Reason And Revolution: Hegel And The Rise Of Social Theory. I can't recall how much, if at all, he gets into the existentialists. For the most part, he's concerned with defending Hegel against British and American philosophers who accuse him of being an inspiration for the fascists.