My shot: The Britain and the US never get involved in WW1, and due to some stupid mistake, France falls early and Russia comes to terms with Germany without being too damaged. However, the economic situation in Europe after the brief European War is very dire, and Russia collapses into a civil war between Republicans and Monarchists. During this war, Muslims within Russia are badly persecuted, and the Porte intervenes, capturing the Caucausus and Central Asia. After Russia recovers and begins its democracication, they begin strongly associating with pan-Slavism, heavily supporting the under-represented (and in some cases opressed) Slavs in Austria-Hungary. A-H eventually collapses, and in its wake Russia extends its sphere of influence into most of the Balkans. This is by 1930, and the Sublime Porte has begun extensive influencing of Persia, Afghanistan, Egypt, and heavy support for the Muslims in British India, British Arabia, and British Africa (Sudan, Somaliland, etc.). With the Porte's support, the British allow their Arabian holdings to become independent with heavy Porte influence. In the 40s, Germany and France get in another war, where they absolutely wreck each other. The Porte steps in and funds rebels in French West Africa, and several said movements succeed. In response, Russia, through un-declared turf wars and diplomacy, ousts the Japanese out of Northern China and Korea, establishing Russian hegemony there. By now (1950) Russia is completely democratic, and the Porte is a constitutional monarchy in all but name. The Porte supports the monarchies of Europe, while Russia allies with the republics in much of the Americas (namely, the US and its sphere of influence in Mesoamerica and the Caribbean). Thus, by 1950, we have a full-blown Cold War between Monarchists and Republics.
Mission accomplished!