Well, I think it's perhaps important to view it in the context of the wider dissolution of the Ottoman Empire in which case the effects were incredibly wide ranging. I can't speak as much to its effects in greater India which is where the majority of the empire's Muslims resided, but the impact here was incredibly influential on the Khilafat movement which was influential not just for Pakistani nationalism but also as an example of an early Islamist movement which, as I understand it, was influential in the thinking Abul Ala Maududi, who was a crucial intellectual of 20th century Islamist movements.

Two things jump to mind though in direct response to your question that I can discuss further. One is the impact that that the Turkish revolution had on the actual borders of the region itself, and in the case of Iraq particularly the northern extent of Mosul province. The Treaty of Sevres envisioned very different borders for the states of the region, and the Turkish revolution negated that.

Secondly, the immediate dissolution of the caliphate briefly led to a race to claim the title by both Sharif Hussein as well as by King Fuad of Egypt. We could debate the extent to which these were "under Britain's domain" at this point, given that Egypt had gained nominal independence in 1922 and Kingdom of Hejaz was not a British mandate, but to a certain extent that's beside the point. The point is that these efforts failed, and did so in such a way that had ramifications for Egypt ruling family (as part of a broad feeling that it lacked legitimacy eventually culminating in the 1952 revolution) as well as the Hashemites (who in losing the Hejaz basically declined to re-take up the Caliphate from Sherif Hussein, though Faisal and Abdullah would continue to rule Transjordan and Iraq as part of British mandates.)

To bring this back to what I said at the top about wider context, it's not the kind of thing that's causal as such but the fact that the Ottoman empire was dissolved and replaced with Ataturk's republic I think has to be borne in mind when considering the intellectual and political developments in the 1920s, 30s, and even 40s, if not longer term, for the entire region. The Caliphal authority of Ottoman Sultans, frankly, was not a crucial, aspect of their rule and rarely had a direct impact on Muslims living in foreign territories. But I think the sort of broad cultural impact of the end of that Caliphate, the end of a central Ottoman authority that had controlled the Arab Middle East for 400 years, the secularizing impulses of elites in many of these countries in the early 20th century, the domination of European colonial powers, all of this has to be borne in mind when you look at, say, the foundation of the Muslim Brotherhood in 1928, or when you look at the career of Izz ad-Din al-Qassem as part of a search for new socio-political-religious structures given the radically different political environment post WWI. Or in the other direction, the seeming positive emulation of what was coming out of Turkey, sometimes in a concrete sense through attempts to carry out similar policies, sometimes only superficially as in the spread of "Effendi culture" in Egypt.