At an elite political level there was a bit of a scramble to claim the Caliphate. Sherif Hussein, the titular and spiritual leader of the Arab revolt, proclaimed himself Caliph in March 1924. King Fuad of Egypt wanted the title for himself and in [edit] 1926 convened an international congress in Cairo in the hope that they would name him Caliph. Unfortunately for Fuad there was no real support for this proposition, either internally or internationally. Sherif Hussein's own position as Caliph would effectively end when the Saudis completed their conquest of the Kingdom of Hejaz, forcing him to abdicate in favor of his eldest son Ali who did not adopt the title, though Hussein's claim to it was so short that it never really enjoyed any support to begin with. The Saudis, fully aware of their lack of Qurayshi lineage, did not seek the Caliphate but wanted to deny to it Hussein which perhaps spurred on their existing conflict with him. Nor did Hussein have the support of his Western allies on the matter. The British, knowing that a majority of the world's Muslims lived within their territory, had no interest in seeing a Caliphate established anywhere as they were fully aware of how perilous it was (or at least seemed) when the Ottoman Sultan called for a Jihad against Britain. The Ottoman Jihad was a bit of a damp squib but it was nonetheless not a chance the British wanted to take.

I'm not familiar with any popular reaction or outrage, or anything like that, but I think you can make a strong case that the dissolution of the Caliphate marked a confusing and distressing turning point in the relation between Islam and modern politics. It is not a coincidence in my mind that the Muslim Brotherhood in Egypt was established in 1928, or that the members of the Khalifat Movement in Pakistan like Abul Ala Maududi would go on to play a key role in the development of Islamist ideology or of the Pakistan movement. Maududi's own writings on what a modern "Islamic State" would look like in large part hinged on the absence of a Caliph.

While I think the actual dissolution of the Caliphate proved to be a relatively minor event in and of itself (the Caliphate was not a particularly important component of the late Ottoman state, and the Ottoman Sultan in any event was not running the country after the Young Turk Revolution), the symbolic understanding by 20th (and 21st) century Muslims that we are now living in a period without a Caliph, a time of Western/outside domination, and of relative cultural/scientific backwardness in contrast with some (relatively abstract) glorious past when we had all those things has played an incredibly important point in political discourse, at least in the Arab world that I'm most familiar with. Reactions to that understanding have been incredibly varied, from Muslim Brotherhood pan-Islamism, to secular pan-Arab nationalism (e.g. Nasserism), down to the contemporary rhetoric of some of the scientific projects of the Gulf states.

Which one?