Hey, this is actually a very good topic.
A few quick thoughts;
Geo-spatially, the following: German domination over North and North-western Europe, satellite states in Vichy France and England, involving some bureaucratic and political-economic friction; Italian domination over South and south-western Europe, and possibly over Northern Africa; Ottoman domination over the Balkan states, with these perhaps eventually disintegrating; and friction with Iran.
Some thoughts on how they would've interacted:
(i) Germany would have created several satellite states in Europe like the Soviet Union; and the Cold War instead would have been fought between the United States and Germany. British control, of course, would immediately have weakened; although if Britain had completely been subsumed under Germany, I doubt they would've allowed Indian independence soon.
(ii) In Asia, Tibet and Manchuria would probably have devolved into puppet states. Japan would certainly logistically not have been able to occupy all of China. If the Azad Hind army had prevailed, as likely would have been the case, it would have unilaterally declared independence and allied itself with Japan, much depending on the interaction between the Azad Hind Army and the Congress. In that situation, two scenarios would prevail: (i) either the subcontinent would have degenerated into many warring factions, eventually splitting and forming several independent, sovereign territories; (ii) Bose would have maintained control with an iron fist, dissipating unrest with terror and turning India into another fascist-state.
(iii) Eventually, my guess is the United States would have prevailed over Germany, for reasons of geo-political and geo-spatial complexity in Europe: Russia is simply too vast to occupy; the United Kingdom was an industrial and entrepreneurial people of its own; and Spain and Portugal were too difficult to govern, , the lack of the same complexities along the U.S.'s contiguous land borders and the eventual adoption of paradoxically different economic systems: the German system was one of top-down bureaucratic-organizational industrie, and would've required several more wars to survive. Although, the Corporatism element of Fascism would have ensured that it prevailed a lot longer than say Communism.
(iv) The global economic situation would probably have been worse, and disparities still wider as state-urged Corporatism took over social-market economics. In the resulting situation, territories with large poor masses like East Asia and the subcontinent would've seen revolutions, that either toppled or were controlled by fascist governments. My guess is: that East-Asia would have seen more Communist revolutions, particularly in the Indonesian archipelago. The UK would of course, have lost all of its Pacific territories.
(v) Central Asia would have remained disintegrated and almost tribalistic; with large German corporations vying for control over its resources.
(vi) Monarchical-japanese culture would probably have dominated East Asia, as it would have dominated trade and industry for a very long time. Perhaps, a level of prosperity, significantly lower than that for the "mainlanders" would have been allowed. Burma and Bangladesh would probably have been very significantly influenced by the Japanese, with Indian culture remaining something of a cross between a proto-Japanese-English-German culture.
The economy, had scenario (ii) in point (ii) prevailed, would have remained early socialist- for population-compelling reasons; eventually degenerating into a corporatist-melange that would've threatened social stability.
(vii) My guess is that the unravellings of this system would begin with the disintegration of the Ottoman empire, with the Middle east degenerating into a cacophony of warring states. Eventually, an uneasy equilibrium would have been maintained between the Ottoman Empire and Iran; with Iran having far more influence in West Asia and Balouchistan-Afghanistan.
