I found this question asked awhile ago but the answers were removed. I find this a very interesting subject and would like to know more, thank you.
There are numerous reasons for the Allies' decision to hand over Constantinople to the Turkish Republic. The most influential reason was the crushing defeat that the Greek Army had suffered in battles such as Dumlupınar (26-30th August). The Turkish Army had retaken Izmir, the 2nd largest populated city in modern Turkey (also home to quite a large Greek/Christian minority) on the 9th September. As a result of the Turkish victories, the Greek Army had to completely abandon their pursuit of the 'Megali Idea' which called for the occupation of Thrace, Ionia (where Izmir/Smyrna was located) and Constantinople. The Greek Army withdrew from Anatolia and suffered some political instability at home, their King Constantine I abdicated and the military took charge of the country. With no Greek military presence in Anatolia it seemed unlikely that the Allies would trust Greece to take ownership of the city. A second reason would be that the leader of the Turkish National Movement, Mustafa Kemal Paşa was offering a peaceful solution to the end of the Turkish War of Independence after the capture of Izmir. In his own words during an interview with the British newspaper, The Daily Mail, Kemal stated "Our demands remain the same after our recent victory as they were before. We ask for Asia Minor, Thrace up to the river Maritsa and Constantinople... We must have our capital and I should in that case be obliged to march on Constantinople with my army, which will be an affair of only a few days. I much prefer to obtain possession by negotiation, though naturally I cannot wait indefinitely." Kemal's willingness to engage in diplomacy had resulted in averting the 1922 Chanak Crisis (a war scare between Turkey and Great Britain) with agreements made at Mudanya and later in 1923, the Treaty of Lausanne was signed giving the Turkish Republic Constantinople and the borders demanded by Kemal in the interview just quoted. As a result of peaceful, yet difficult negotiations at Lausanne between the Allies and Turkey (represented by Ismet Inönü) Constaninople became a part of the Republic of Turkey in October 1923. The third reason would be that the Pro-Greek nature of the Allies had all but evaporated by 1922 , British PM Lloyd George was ousted due to the Chanak Crisis and replaced by the Pro-Turk Bonar Law of the Conservative Party. The French had also reached a seperate agreement with Turkey after their conflict in the south-west of anatolia, on the border with modern-day Syria, was resolved in 1921. All of these factors had shown Kemal's forces were those to be reckoned with and that the Greek victory many foresaw as inevitable was, by 1922, completely out of the picture. As a result, by giving the Turks Constantinople the Allies would have avoided a general conflict that would have been costly, and were allowed to keep favourable relations with Mustafa Kemal and the Turkish Republic as Kemal was not vehemently Anti-Western. For a better in-depth view of the battles and commanders of the Turkish War of Independence and the negotiations that followed i would suggest Andrew Mango's book 'Atatürk', i have been re-reading lately and the detail is astonishing.