In my challenge the proposed nation is an independent nation state which comprises only the area specified and is not part of a larger nation.
Possible scenario: POD is the Balkan War of 1912 to 1913 which goes worse for the Ottomans than in OTL and they lose all their territory in Europe including Constantinople, which is annexed by Greece.
First World War happens as in OTL, though with Greece holding Constantinople I don't know if the Gallipolli campaign would have happened.
Post war settlement is as in OTL, with Greece confirmed in its pre war boundaries. Romania annexes Bessarabia. In the Graeco-Turkish War of 1919-1923 Turkey does not regain its European territories.
Crown Prince George becomes King of Greece as George II in 1922. He is not forced into exile in December 1923 as in OTL. In 1924 he moves his capital to Constantinople and declares the restoration of the Byzantine Empire and assumes the title of Emperor George I.
Sometime in the early 1930s the Byzantine Empire invades and annexes Albania. An attempt by Italian troops to invade Albania in April 1939 is decisively repulsed.
The Second World War happens as in OTL up to April 1941 when German and Italian forces invade the Empire. They advance rapidly and conquer OTL Albania and Greece. Winston Churchill decides to make a determined effort not to let Constantinople and the surrounding area fall into enemy hands. Constantinople is declared an open city like Paris and Rome were in OTL.
German and Italian troops cross the Maritsa river (
http://encarta.msn.com/map_701514500/maritsa.html ) in late April 1941. They capture Adrianople (Edirne) after heavy fighting. But dogged resistance by British and Byzantine forces means that German progress is slow. Hitler invades the Soviet Union as in OTL and withdraws his troops from the Byzantine front. The Maritsa becomes the de facto boundary between the Free and German/Italian-occupied Byzantine Empire.
In October 1943 Byzantine and British forces cross the Maritsa. By April 1944 they have liberated all of the Empire and are advancing slowly against heavy German resistance through the Balkans. Because Germany is also fighting against the Soviet Union in eastern Europe, the Soviet advance is quicker than in OTL. Germany surrenders in late March 1945 and the war in Europe is over
The route of the advance through the Balkans the Byzantine forces passes through Bulgaria, Romania and Yugoslavia. The rulers of Bulgaria and Romania consent to their countries becoming provinces of the Imperial Federation of the Byzantium. A vicious civil war in OTL Yugoslavia between the Communists under Tito and the Byzantine/anti-Communist alliance ends with the victory of the latter in June 1946. Yugoslavia joins the Empire with its different provinces - Croatia, Serbia etc as provinces of the Imperial Federation.
Byzantium is now a prosperous stable democracy. The Emperor rules as a constitutional monarch.