Just to help, which Caliphate are you referring to? The Umayyad Caliphate, Abbasid Caliphate, Fatimid Caliphate, etc? There wasn't one caliphate and often times there were two simultaneously, such as the Umayyads, who ruled mostly over Spain, and the Abbasids, who ruled mostly over traditional Arabia and the Levant. Which one do you have in mind?

There was discussion after WW1 about creating a new caliphate but there was no consensus about who who should assume the title. The Hashemites, who had been promised most of the Arab world in the Hussein-McMahon correspondences, had wanted the title for the Sharif of Mecca, King Hussein bin Ali who actually assumed it, briefly. The final extent of their domain was limited to Transjordan, Iraq, and the Hedjaz, and the Hedjaz only briefly because that was conquered by the Al Saud soon after, who also thought they might have a claim. King Fuad of Egypt, and later his son Farouk also had pretensions to the title but these were widely opposed. Fuad and Farouk both had a rather notorious reputation for un-Islamic behavior. The Turks, who formerly held the title, wanted a secular state and had abolished it.

All of these were under the sphere of influence, if not the direct control, of the British Empire who, through their imperial holdings in Greater India (i.e. Pakistan and Bangladesh as well) actually had a larger number of Muslims in their domain than did the old Ottoman empire, and certainly more than any of the post-Ottoman states. As such they had no desire to see the caliphate restored and this whole series of events was something of a coup for them.

The pan-Arab Socialist movements that eventually toppled many of these states, for example in Iraq, and Egypt, were secular and had no interest in the Caliphate.

I've read some much longer sources on the politics of this but this this seems to sum it up in only a few pages: http://books.google.co.uk/books?id=ZAXNxxkJKYsC&pg=PA301&lpg=PA301&dq=sharif+hussein+king+fuad+caliphate&source=bl&ots=blVhH5azwb&sig=EKMuhRK1uSwtsD6zCvcBdgfA0I4&hl=en&sa=X&ei=8uNYU9bnCNPRsATrn4GgCg&ved=0CD0Q6AEwAw#v=onepage&q=sharif%20hussein%20king%20fuad%20caliphate&f=false