As others have stated, Goring (or an army junta) would be the obvious successors to Hitler. It is worthy to note that at this time, the anti-Nazi generals had a rather well geared coup plot ready to spring up if Hitler had given order to invade Czechoslovakia, so they could easily seize power if Germany looks like any slipping into chaos. OTOH, Goring was the by far most influential Nazi bigwig after Hitler, a foreign policy moderate as Nazis went, and somewhat popular with the army, so in the wake of a terrorist attack on the Fatherland, the generals may back up Goring instead, esp. if he gives timely guarantees of a moderate foreign policy (so no order for invasion, the generals would not back it up without guarantees that UK and France approve it). So in my reasoned opinion it is a coin toss between Hermann and an Heer junta.
Now, as far as the new foreign policy of Germany goes, there are not going to be substantial differences either way: Goring was a neo-Wilhelmine moderate that supported retaking the Sudetenland and the territories lost to Poland in 1919 but had no interest in Lebensraum and was strongly adverse to war with the Western powers. As for the generals, they strongly support retaking Sudetenland and as closer to the 1914 border as possible, but they don't want any war with the West and would deem Germany a satisfied power after all the irredentist claims are satisfied.
Given that the bombing is going foster some solidarity between the four powers, I expect a German-Western detente and confirmation of appeasement to be in the wings after the new leaders are stabilized.
As it concerns Italy, given that Mussolini and Ciano are dead, the most likely successor are either an army junta, if the king and the generals decice that the fascist regime has outlived its usefulness, or alternatively Balbo looks like the most plausible successor, he was charismatic, rspected by the army, and popular. The army would almost surely suppress any attempt by Farinacci and the radicals.
In both cases, the new Italian leadership would have been less enthusiastic to go along with Hitler's adventures, but they are going to find no big problem to align themselves with a moderately irredentist/expansionist German policy that strives to satisfy its objectives without a general war. It's pretty much the same thing they would want to do themselves. So Italy shall maintain the Axis and keep a less bombastic foreign policy while looking for the good opportunity to expand in the Western Balkans, grabbing Albania and destroying Yugoslavia in combination with Hungary and Bulgaria. They are going to leave Greece alone, however, too dear to Britain.
As it concerns Britain and France, I am not so familiar as to name a successor, but I would expect that someone would be named that would continue the appeasement policy. In all likelihood, a different party than Germany or Italy would be blamed (quite proabably Czech nationalists and/or the Communists) so Anglo-French leadership would see this terrorist attack as a confirmation of the munich policy, and the moderate course of the new German & Italian leaderships would confirm this.
As I said, it is most likely that Prague, Moscow, or the Comintern at large get blamed. So after the political chaos gets settled, the Munich conference would reconvene and the cession of the Sudetenland would get reaffirmed even more vigorously.
What happens to Europe essentially depends on what Stalin does, whether he thinks of exploiting the perceived weakness of the capitalist powers (or pre-empting their anti-communist alliance) or gets cowed.
Scenario 1:
If Stalin is cowed by the European united front, Germany gets the Sudetenland and leaves rump Czechia alone. Britain and France are pleased with Germany keeping its word and deem the appeasement strategy and success. So when Germany makes its last claims in Europe about Poland, they make Munich II and back Berlin's claims on the Corridor and Upper Silesia. Poland is alone and it is a coin's toss whether they give in or go into a kamizake lone fight with Germany, but Paris and London shake their head at Pole stubborness and look the other way in the latter case. Germany grabs the 1914 border and sets up Poland as a satellite, which the Western powers find acceptable.
In the meanwhile, Italy exploits the situation to annex Albania and pick a fight with Yugoslavia in combination with Hungary and Bulgaria with Croat nationalist skirmishes as the casus belli. Paris and London shake their heads but they aren't going to spill their blood to save Serbia's little empire, the British could not care less and for France the usefulness of Yugoslavia as a client has expired with appeasement. It takes some time for Italy to win but the rebellion of Slovenia and Croatia seals the doom of Yugoslavia. Italy annexes Dalmatia and Montenegro and sets up Slovenia and Croatia as clients, maybe grabbing a piece of Slovenia. Germany quietly supports and maybe grabs a piece of Slovenia, too. Hungary reannexes Vojvodina and Bulgaria grabs Vardar Macedonia.
Germany and Italy become satisfied powers, and gradually build a strong detente with the Western powers, Stalin does not dare to do anything against a united Europe and WWII in Europe is butterflied away. At some point a war between Hungary and Romania for Transylvania occurs, the outcome is not certain but most likely Hungary recovers northern Transylvania. The Spanish Civil War goes the same way as OTL.
At some point a war between Japan and Russia is possible, which ends up in a stalemate, Russia grabs Manchuria and Japan grabs Sakhalin. Japan is unlikely to do anything in South East Asia but keeps bleeding in China until it realizes it can't win. Unless Russia intevenes to back Chiang and/or Mao, which could bring a detente between Britain, America, and Japan as they join hands to keep Russia out of China. Not a direct intervention, likely, but strong support to whatever local anti-Communist proxy in China they can find. Perhaps trying to steer Chiang back to anti-Communism in exhcangw with Japanese withdrawl. Difficult to foresee China's ultimate fate, but some kind of division between Red and KMT China seems the most likely outcome.
The world remains a multipolar imperialistic place with America, Russia, Britain, France, Germany, Italy, and Japan all worthy greeat powers vying for influence as decolonization veerryy slow happens and fascism and communism remain respectable ideologies.