I don't see this happening without the intervention of what I have come to learn the forum calls ASBs.
The Axis had enough trouble in Northern Africa as it was. Altered circumstances, and luck, MIGHT have resulted in an Axis victory in Northern Africa, and occupation of North Africa and the Palestine Mandate. However, Hitler did not have the ressources to go all the way to mo-fo'ing India

To do that, he'd have to successfully invade, and hold, Iraq and Persia, which is a lot LOT harder than it sounds writing it here.
To push to India (for what purpose, anyway?), the Wehrmacht would need to occupy all of the area between Mandate Palestine and India. As said, that means invading and occupying Iraq and Iran successfully. Now, even if we place this push towards India during the phase where Iraq was a de facto Axis member, thus eliminating the need for an invasion and occupation of that country, they'd still need to invade and occupy Persia, and hold it successfully in order for their supply lines to allow for an invasion of India. This would mean a considerable drain on manpower, not even going into the fact that the German forces weren't at all used to the extreme heat that can be Iraq and Iran in the summer.
Let's say they pull that off, though. Let's say that the Wehrmacht has conquered Persia (securing the oil fields in the process, btw, but this is too little payoff at too high a price). Now they stand at the borders of India.
Why would they want to invade India in the first place? Yeah, sticking it to the Brits by taking the jewel in the crown of the Empire would make a good laugh in the Führerbunker, but I see little strategic and ideological reason for the nazis to do this.
And anyway, a Middle Eastern campaign this extensive, will 100% assuredly put Barbarossa on indefinite hold. Germany WAS powerful, but it was not powerful enough to carry out two extensive invasions across two different theaters.
The main objective was Lebensraum in the east - east meaning Poland and the USSR - and there was simply no reason to bother with the rest of the Middle East, once Northern Africa, Cyprus, Malta, Crete, and Mandate Palestine had been secured.
In a game of Hearts of Iron, I believe I DID do an Axis invasion of India once.
But:
1: What's possible in a computer game is a far cry from what's possible in the real world
2: I cheated, giving myself infinite ressources
tl;dr: Can't be done without ASBs