The Nazi reaction was fairly delusional. A day after Roosevelt's death, Hitler issued a proclamation to the soldiers fighting on the Eastern Front which included:

In this hour, the entire German Volk looks to you, my fighters in the east, and hopes that, through your steadfastness, zeal, arms, and leadership, the Bolshevik attack will drown in a bloodbath. At this moment, in which Providence has removed the greatest war criminal of all time from this earth, the turning point of this war is being decided.

This fit into a narrative that Roosevelt, who had pushed at Casablanca for the "Unconditional Surrender" of the Axis, was what stood in the way of a negotiated peace, and more importantly, the Western Allies coming to realize that they needed to ally with Germany to fight the real enemy: Bolsheviks. Hitler was not alone in such expression. A few days later, Goebbels gave his traditional radio address on the eve of Hitler's birthday and noted:

The head of the enemy conspiracy has been smashed by Fate. It is this same Fate that allowed the Führer to stand upright and without any injuries among the dead, the seriously injured, and the ruins of July 20, so that he would be able to conclude his work.

Two days later of course, Hitler would be woken by the first Soviet shells hitting the city, although his public comments continued to push the notion of ultimate victory still to come when he phoned Gen. Koller:

You will see, the Russians will suffer their greatest defeat, the bloodiest defeat in their history, in front of the gates of Berlin.

In many ways it was all a front for him though, with Hitler swinging between manic and depressive moods, and at several points over the next few days he discussed suicide as the war was lost. Several generals who met with him on the 22nd related to their subordinates similar sentiments as that given by Jodl when he noted:

Hitler has given up. He has decided to stay in Berlin, lead its defense, and shoot himself at the last moment.

Goebbels though perhaps internalized it more. Although certainly with Hitler's declaration to remain in Berlin and die Goebbels declared his intent to follow Hitler to hell, he nevertheless was considerably more bouyant in his own reactions. While with Hitler it seems to mostly mask an internal depression about the nearing doom, this reads as less true for the Propaganda Minister, who perhaps took too much of his own medicine. Not that he too didn't have moments of doubt which he committed to his diary, but upon hearing the news, a secretary with him at the time related that he declared:

Now bring out our best champagne and let us have a telephone talk with the Fuehrer.

In that conversation shortly after, with his mood described as being "in an ecstasy" he told Hitler:

My Fuehrer, I congratulate you! Roosevelt is dead. It is written in the stars that the second half of April will be the turning-point for us. This is Friday, April the 13th. It is the turning-point!"

Others similarly saw providence at work. Schwerin von Krosigk, the Finance Minister relating his hearing the news in his diary, writing about how a discussion on the dwindling ammunition supply was interrupted by a fateful call:

Would this really be the end which my reason had so long seen as inevitable, but which my spirit had striven so hard not to see? At that moment the telephone rang; the state secretary wished to speak to me. What could he want at so late an hour? He only uttered one short sentence: "Roosevelt is dead." We felt the wings of the Angel of History rustle through the room. Could this be the long-desired change of fortune?

Reaching out to Goebbels to discuss how best to reap this supposed coup, Goebbels equated Roosevelt's death to the "Miracle of the House of Brandenburg" which had seen the Russo-Austrian alliance fail to crush Prussia following the death of Empress Elizabeth, Krosigk recording the conversation thus:

[Goebbels] had developed his thesis that, for reasons of Historical Necessity and Justice, a change of fortune was inevitable, like the Miracle of the House of Brandenburg in the Seven Years War. One of the staff officers had somewhat sceptically and ironically asked, What Czarina will die this time? That, Goebbels had replied, he could not say; but Fate still held many possibilities in her hand. Then he had driven home, and had heard the news of Roosevelt's death. Immediately, he had telephoned to Busse, and said, "The Czarina is dead". Busse had told him that this made a great impression on his soldiers; now they saw another chance. Goebbels believed that this news would awaken a new spirit of hope in the whole German people; they would surely see in it the power of Historical Necessity and Justice .... I [says the sententious Count] interrupted, "Say rather, of God" ....

To return to the beginning though, this was all delusional fantasy. The image of Roosevelt as a warmonger, keeping the American people in a war they didn't want to be part of and champing at the bit to pull out, was a necessary component of these reactions, and also utterly divorced from reality, even if the war wasn't already on the edge of victory. Likewise without Roosevelt propping everything up, the British too would seek peace. However much propaganda the Nazis might spew out about how "the political crisis in the enemy grows daily" that of course didn't make it true

Sources

Domarus, Max. The Complete Hitler: A Digital Desktop Reference to His Speeches and Proclamations 1932-1945. Bolchazy-Carducci Publishers, 2007.

Eberle, Henrik & Matthias Uhl (eds). The Hitler Book: The Secret Dossier Prepared for Stalin from the Interrogations of Hitler's Personal Aides. PublicAffairs, 2005.

Kershaw, Ian. The End: The Defiance and Destruction of Hitler's Germany, 1944-45. Penguin, 2012.

Trevor-Roper, Hugh. The Last Days of Hitler PalgraveMacMillan, 1995.

Weinberg, Gerhard L. Germany, Hitler, and World War II. Cambridge University Press, 1995.