What was the psychological/theological impact on devout Muslims worldwide when Ataturk legally dissolved the Ottoman Caliphate in 1924? Was there outrage? Mourning?

The Muslims of India certainly felt shocked and betrayed at the dissolving of the Caliphate. They had been pleading with the British not to dismantle the Caliphate. Indian Muslims had been sending medical missions to the Ottoman Empire in order to help their war efforts during the Balkan Wars, after World War I, and during the Turkish War of Independence, in the form of financial help, food, clothing, etc. to the Turks (See: Khilafat Movement). So that feeling betrayal and disappointment they felt when Ataturk abolished the Caliph was warranted from their perspective. The dismantling also gave strength to the Pakistan movement among members of the Aligarh Muslim intellectuals, but it continued to be opposed by members of Jamiat al-Ulema e-Hind, which was mostly composed of traditional scholars and opposed partition.

Meanwhile, the British had originally promised that the Caliphate would remain untouched, but were secretly preparing to set up the Hashemites, in particular the Sherif of Mecca as the new Caliph. However, they failed to realize that he did not have the same name recognition that the Ottomans had, not to mention the history and tradition, and many felt that he would be nothing more but a Western puppet. Not to mention, he had lost the Kingdom of Hejaz, the home of the holy cities of Mecca and Medina, to Ibn Saud who subsequently went on to establish the Kingdom of Saudi Arabia a few years later in 1932.

The loss of the Caliph was certainly a shock to many Muslims in the Arab world, Central Asia, South and Southeast Asia. Even among the Turks, there were quite a few lamentations as the Caliph continued to be a popular religious figure, but the traditional order was too weak to do anything about it. There was a conference held in 1926 in Cairo to decide what should be done in order to choose a new Caliph. However, a new Caliph wasn't chosen because there wasn't enough representation of Muslims from around the world and no consensus was reached. The conference expected to come back the following year to decide but it never took place, as this was the era where we began to see a rise in pseudo-secular/secular nationalist movements throughout the Muslim world, particularly in places like India, Afghanistan, Indonesia, Egypt, Iraq, Algeria, etc. All sorts of anti-colonial/anti-Imperialist movements were sprouting up throughout the Muslim world, especially in the Middle East.

Areas not affected by the loss of the Caliph

The loss of the Caliph did not have major repercussions in the Western part of North Africa and the region of northern Nigeria and Niger. The Maghreb (in particular Morocco) never really recognized the Ottoman Caliph, in fact, they didn't recognize the Caliph since the Abbasid took power in Baghdad. They declared their allegiance to the Umayyads in Spain. The argument they hold is that if there was to be the establishment of the Caliph, the best likely candidate would have to be the King of Morocco, but outside of Morocco and parts of Western Africa, no one really accepts this notion.

In areas around Nigeria, they already have a Caliph, who is the descendant of the Sokoto Caliphate. Again, outside of parts of West Africa, the Caliph isn't recognized by too many Muslims.

Source(s):

A Peace to End All Peace: The Fall of the Ottoman Empire and the Creation of the Modern Middle East by David Fromkin

Islamic Movements in the Arab World 1913–1966

I'll add some more when I get a chance

Edit: Grammar