Hitler's will and testament split up the position of Führer, the single unitary leader of Germany, back into their original posts of Chancellor and President. Before his suicide, Hitler had designated Goebbels as his successor at the Chancellery and the chief of the Kriegsmarine, Karl Dönitz, as President.

The rationale behind Goebbels's succession was somewhat obvious. Unlike other members of Hitler's inner circle, Goebbels remained outwardly loyal to Hitler. Both Himmler and Göring had proven somewhat disloyal in the last weeks of Hitler's life. Göring had requested clarification on whether or not he should take the reins of power since Hitler was about to be surrounded in Berlin. Martin Bormann twisted this request into an appearance that the Luftwaffe chief was trying to usurp Hitler's authority. This led to Hitler's cashiering of Göring and his replacement by the far more pliable Ritter von Greim on 26 April. Hitler's row with Göring was part due to Göring's own actions, but also Bormann's machinations. Himmler's betrayal of Hitler was much more clear-cut. The SS chief thought himself to be the natural successor for Hitler and entertained notions of negotiating with the Western Allies as the head of a new German government. The Führerbunker received word via a Reuters radio report concurrent to Göring's telegram on his position that Himmler was attempting to negotiate with the Allies. Himmler had made overtures on 20-23 April to the Swedish envoy Count Bernadotte to set up meetings with the Western Allies. These overtures went nowhere but they were enough for Hitler to order Himmler's arrest for treason. The failure of SS general Felix Steiner to relieve Berlin and the perceived disobedience of the SS under Sepp Dietrich's command also fit into a pattern for Hitler of the base treason of the SS against him. Many witnesses from the Führerbunker attested that Hitler's outbursts when he heard news of Himmler's betrayal were among some of the most apoplectic Hitler had ever given, which says a lot. So handing off part of his succession to Goebbels became logical as the Propaganda Minister had remained in Berlin and shown no active signs of disloyalty.

Dönitz too had also proven his loyalty in Hitler's last days. The Kriegsmarine had sent a number of shore troops to relieve Berlin and while this did not alter the battle one bit, it was a more active defense of Germany in Hitler's eyes than Steiner's actions. Dönitz received word of his promotion with a degree of equanimity, but like Himmler, who was also present in his entourage despite Hitler's formal sanctioning him for treason and expelling him from the party, Dönitz entertained notions of a separate peace with the Western Allies. The new German head of state sent out reports of the threat of Bolshevism towards Germany and tried to evacuate as many German troops from the collapsed Eastern front as possible. Dönitz also approved of a number of capitulations to the Western Allies such as the surrender of NW Germany, Bavaria, the Netherlands, and Denmark in 4 May. The diplomacy the new German government conducted in week between Hitler's suicide and the surrender at Rheims on 7 May though showed that a partial surrender was impossible and the Allies were united in wanting a thorough end to the war.

This all meant that Dönitz government was on borrowed time. The various Allied governments did have their disagreements and tensions, but they were united on the front that the Nazi party and its officialdom were not reliable partners in the occupation. Dönitz had a use for the Allies in that he provided a central political voice which could facilitate a unilateral German capitulation. But he was useless to them after that aim had been achieved. Dönitz tried to insert himself as an intermediary to the newly-formed Allied Control Commission governing Germany after 7 May, but he was rebuffed each time. The Commission formally dissolved the Dönitz "government" on 23 May 1945.

Circling back to the OP's question, the answer to who led Germany after 30 April is really no one was running the country. Dönitz had enough vestigial authority to communicate a general surrender, but his ability to do so was also greatly facilitated by the Allies' desire for a formal German military capitulation. But the situation on the ground beyond Dönitz's headquarters in Flensburg was chaotic. German troops were deserting in massive numbers and one of the common features in memoir literature from the period of collapse is many cities' and towns' party leaders and civic authorities simply absconded from their duties when Allied forces approached. Dönitz could authorize surrenders, but there was precious little he could do besides this. More often than not, local and municipal administration were either left to their own devices if not under direct Allied occupation or rump administrations worked with the military governments.

This state of affairs was a byproduct of Hitler's own careless methods of rule and governance. As his haphazard delegation of Chancellor and President shows, Hitler did not really give much thought to the practicalities of running a complex state and still less consideration to what would happen after him. The splitting of the office of Führer had no real legal precedent. Hitler also knew Goebbels would probably follow his example and commit suicide, as Goebbels did 24 hours after Hitler's death when the Soviets rebuffed his negotiations, but nonetheless designated him as Chancellor. It was an irresponsible political act, but nonetheless one that was totally in keeping with Hitler's governing style.