British played the main role, and in 1922 practically ordered the Greek government to hand over Constantinoupolis and Eastern Thrace to the Turks. However, the main force behind this was France who demanded that Turkey should be rewarded for being a friend of France. Italy played a role as well.

Some outline of the events is at http://www.greek-genocide.net/index.php/overview/documentation/130-eastern-thrace

The main act of genocide in Asia Minor was orchestrated by the Germans, who sent their military advisors to suggest that Turks systematically genocide the Christian population. Which is why Hitler admired Kemal so much and tried to do the same against Poland. The man who organized the genocide was the Jewish-German Liman von Sanders

The British continued to play a similar role during after WWII as well. They funded the communists and orchestrated the Greek Civil War so that Greece wouldn't have the opportunity to demand the liberation of Cyprus. Finally, they preferred to bring Turkey into Cyprus than having Greece have it.

The whole British involvement in the massacres of Greek Civil War by the communists has been documented in this book