I thought the issue in question was Islam today, just with a POD before 1900. Removing them from power could be such if they were colonised in the 19th century, following a more aggressive imperialism or an earlier collapse ot the Ottoman Empire. If I misunderstood, I apologise.
You raise an interesting point, although your comment, as I interpreted it, suggested that Islam's current strength and diffusion is due to the current exploitation of oil, which is just strange.
The point is that there are two potential understandings of the term POD - one that emphasizes the cause, and the other the effect. You didn't really explain how Arab and Iranian Muslims would be "removed" from ruling oil-rich areas. Does the oil disappear? Is it never discovered? Does someone else take it over (I once toyed with the idea of a TL featuring a Mandaean and Zoroastrian "Kingdom of Ormus," ruled first by the Portuguese, and later by the British, that would encompass most of the islands in the Persian Gulf)? In the latter case, it would require substantial changes to the OTL, beginning most likely in the 16th century - but the other cases would produce a TL that was remarkably similar to ours before the 20th century. Thus, the POD for the first would be prehistoric (but without any overt divergence until the 20th century), the POD for the second in the 20th century, and the POD for the third sometime during the 16th century or thereabouts.
In any case, the specific effect that you are talking about (lack of oil money for Islamic states like Saudi Arabia and post-1979 Iran) clearly only comes into play in the latter half of the 20th century. That would be when the TL really diverges from our own, unless we're talking about a TL that was already radically different, in which case it's pointless to talk about oil as that would be a rather minor difference only coming into play centuries down the line.