How and why the Nazis win, and what victory really means in such circumstances are necessarily to determining the disposition of the British Empire, and the colonies of France as well. Italy's too for that matter.
This. We cannot have a “aftermath of Nazi victory” discussion without knowing the form that victory took.
Let's imagine Lord Halifax becomes Prime Minister of the United Kingdom (a cliched scenario, I know) and pursues an armistice with Germany after the BEF is destroyed at Dunkirk and France's defenses collapse as in OTL. A peace treaty is developed and signed in summer 1940, ending the war in (western) Europe.
Under such a treaty, I doubt the British would lose any of their colonies. They may have lost the land war in Europe, but their navy would still rule the seas and could ensure the safety of the colonies. At the most, Britain might return former German colonies in Africa like Cameroon, Tanzania, and Togo (Namibia, which by then was administered by South Africa, probably would not be given up). But I think it's unlikely Britain would surrender any more territories than those. And under no circumstances would the British Isles be occupied by German troops or puppetized.
France, now under Petain's regime, probably retains her colonial empire. At most, the French might return former German colonies like their portions of Cameroon and Togo. Perhaps Madagascar would be given to Germany if the Nazis were serious about implementing the Madagascar Plan (which I think was infeasible because the Germans lacked the ships to transport millions of Jews overseas, and so the OTL Final Solution would still emerge).
Denmark might retain Iceland and Greenland if its independence is preserved (albeit as a German puppet state). But the British would refuse to let the Germans step foot in Iceland and Greenland, since that would allow Germany to more easily project power in the North Atlantic and threaten the sea lanes between North America and the British Isles. So I think the 1940 peace treaty would either declare that Iceland and Greenland are neutral territories, or grant independence to Iceland (which falls into the British sphere) while Greenland becomes a British protectorate.
I'm less sure about the fates of Belgian and Dutch colonies. If Belgium and the Netherlands end up like Denmark, being nominally independent but still occupied by German troops and subject to German influence, then they might keep all of their colonies. If Belgium and Netherlands end up like in OTL, being placed under direct German administration, then the colonies might pledge their allegiance to the governments-in-exile and fall under British (and later American) control.
The peace treaty might give Germany access to the markets of the French, Belgian, and Dutch colonies. That could provide the Germans with some of the resources they need for their war effort, especially once the inevitable war with the Soviet Union erupts.
If Germany has access to the resources of the Belgian Congo, then they'll have access to the uranium from Shinkolobwe. That would boost their nuclear weapon project, although it would still be plagued by other issues and would be many years away from producing a working device.
In Southeast Asia, Japan might be deterred from attacking and seizing European colonies in the region if Britain is at peace and French Indochina and the Dutch East Indies are still controlled by their parent governments in Europe. Britain would have more troops, ships, and aircraft to commit to its colonies in the region - and I'm sure they would take measures to deter Japanese aggression.
If Indonesia remains under Dutch control (and is indirectly under German control), then I wonder if Japan might try to seek a trade agreement that gives them access to more oil, rubber, and other resources from there. Perhaps Germany could pressure the Dutch to give the Japanese a favorable deal. If Japan can secure an alternate source of oil, then once America eventually embargoes them for their actions in Asia*, Japan will still have a backup option to fuel itself. Thus, there may not be a war between Japan and the United States (not yet, at least).
*The OTL oil embargo was triggered when Japan occupied southern French Indochina, violating the agreement they made with the Vichy regime promising they would only occupy the north of the colony. The occupation of the south was seen as prelude to Japanese aggression against the rest of Southeast Asia, and so the Allies cut Japan off from their oil. If Japan is deterred from making moves against the rest of the colonies and/or can peacefully acquire more resources from the Dutch East Indies through a trade agreement, then they might not occupy southern French Indochina in the first place. But America would probably find a reason to embargo Japan eventually, since Japan would still be brutalizing China.
I'm not sure what happens to French or Dutch colonies in the Western Hemisphere, such as their islands in the Caribbean and Dutch and French Guiana. I don't know if the British and/or Americans would seize the colonies, or if they would be granted independence, or if they would remain under French and Dutch control (even if both of those countries become subservient to the Nazis). Surely the United States would refuse to allow the Axis to have any territorial presence in the Western Hemisphere, so I'm tempted to say they would join with the British to capture those colonies.