It was commonly referred to as Turkey after WWI, and pretty much under the control of Mustafa Kemal and the Turkish Grand National Assembly from mid-1919 onward.

The Ottoman government was nominally still in control when the war ended, but the military was under the command of Mustafa Kemal (later Ataturk), who was unhappy with the Ottoman government in Istanbul, in particular because the Ottoman government so easily (in his view) accepted the partition and occupation of Anatolia by the allies.

This is a long story, but the simple version is that by mid summer 1919, Kemal's group and other resistors had set up their base of operations in Ankara. The Ottoman delegation's acceptance of the terms imposed on them at Sevres turned the nationalists entirely against the Ottoman administration. The military followed Kemal and gave their allegiance to Ankara and vowed to fight the partition of Anatolia.

Under different circumstances, this might have been considered a civil war, but, while there was some fighting, the Ottoman government was, essentially, rendered irrelevant by early 1920 when the Turkish Grand National Assembly was convened in Ankara--the Ottomans were, essentially, a paper tiger while the Ankara-based nationalist government were the ones who engaged upon what is now known as the war of Independence, driving the Greek armies out of the Aegean costal areas, and the Soviet and Armenian armies out of the provinces in the east.

When the Nationalists were able to force a renegotiation of the Treaty of Sevres--at Lausanne--it was they, not the Ottomans, who represented and negotiated the Turkish position.

The abolition of the Ottoman Empire by the Grand National Assembly was the first step in formalizing the transition of power, but the same provisional government of the Turkish nationalists had been running things from Ankara since 1919; the "declaration of the Republic" was, in essence, a formality recognizing themselves as the sole legitimate government and a declaration of a clean break from the past.

But, if one looks at newspapers from this time period, you will see that the country is referred to as "Turkey" throughout.