Abdul Mejid II was the last caliph. And on the contrary , to the OP's assumption, he was deposed because he was too much under influence of parties holding his strings. This New York Times article from 1924 puts light.
《Sorry, i couldn't shake $4 for the whole article》
Another closer is excerpt from a New York Times article, April 13 ,1920.
Caliph was unpopular in Turkey, because they saw it a tool in hands of west and imposing on their nationalism. Ironically, very nature of Islam, does not provide for nationalism. And being immediately close to him, they could oust the Caliph.
But, puppet or not, he was still considered by Muslims of the world as their religious ruler. And they were quite offended by him being toyed around. An example, is The Khilafat Movement in India incepted by brothers Mohammed Alu and Shaukat Ali. This became into a vigorous movement. Gandhiji himself was elected president of All India Khilafat Committe. Infact, it was integrated into the Non cooperation Movement, one of the declared aims of the which, was "RESTORATION OF THE POSITION OF CALIPH"
But, after he was deposed, well, nothing much could be done, as most of the Muslim world, did not have a strong sovereign exsistence to weild enough influence.