At the outset of the war, Germany and Italy had separate goals. The German goal was to conquer vast swaths of Eurasia and forcibly depopulate it in order to create lebensraum for German settlers. The Italian goal was to establish hegemony over the Mediterranean Sea.
Germany would thus have virtually zero interest in either Africa or the Middle East, except insofar as they may have served the "depopulation" goal. (For example, there was an official German plan to forcibly relocate Jews to Madagascar.) Italy would want to seize territory in the Levant and North Africa, and potentially expand their African possessions.
France held, at the beginning of the war, what is today Syria and Lebanon along with much of sub-Saharan Africa. Under the terms of their 1940 armistice, they were even allowed to keep all of this territory, even though they were forced to give up more valuable territory such as Paris. So it's hard to imagine any scenario where Italy would expand at France's expense. Furthermore, much to the dismay of Charles de Gaulle, the French colonial governments in these areas did not, by and large, see any reason to break with Vichy.
Spain and Portugal were both neutral, and Spain had an Axis-aligned government, so I don't see their territory being affected much. The Belgian Congo is a puzzling case. Belgium itself wouldn't exist, but it's not entirely clear that Germany would want to keep the Congo. It probably goes to either Italy or Vichy France.
That leaves the UK. Early in the war, Hitler was amenable to making peace with the UK and allowing them to keep their Empire. If, early in the war, Churchill's government had collapsed and replaced with a defeatist government as happened in France, there would be a significant possibility of Britain leaving the war intact and in control of their African and Middle Eastern possessions, colonies, dominions, and mandates. However, the longer the war dragged on, the more Germany would have demanded in concessions. Furthermore, Italy's plans for the Mediterranean could not easily coexist with British control of Palestine, Egypt (and hence the Suez Canal), Cyprus, Malta, and Gibraltar.
Another thing to consider is what set of counterfactual circumstances would have led to Axis victory. This establishes both facts-on-the ground and, potentially, changes the set of interested parties. To name some examples:
- If Italy were, in general, more competent than IOTL, they would end up in a much better negotiating position and may have been able to forcibly seize some or all of Egypt, Palestine, Transjordan, or more of British Africa, supporting their desire to retain these territories.
- If Spain joined the war, their obvious first move would have been to seize Gibraltar. Presumably, they'd want to keep it.
- If Vichy France joined the war, they might have been interested in some British colonial possessions, along with the Belgian Congo. They would also have had more influence in the ultimate peace treaty, curbing Spanish designs on French Morocco or Italian designs on French Syria or Tunisia or Algeria.
- Most British colonies had at least one native movement that at least somewhat entertained the notion of collaborating with the Axis. In Iraq, that metastasized into a coup (which the British promptly quashed). If one or more of those movements managed to hold effective control of the country at the end of the war, they would presumably expect the Axis to recognize the legitimacy of their government, and the Axis would have some vested interest in doing so.
- If Britain were subverted by defeatists and collaborationists as France was, they may have even been able to resurrect some style of "friendly" relations with Germany and retain many of their possessions, much to Italy's chagrin. This was vaguely the goal of the Rudolf Hess plot in 1941.