I don't know much about the Great Game itself, but might this turn Russia's attention away from the Balkans and perhaps water down Pan-Slavism? If Russia's real interests lie in exploiting their Great Game victory in Central Asia, they might feel less of a need to get involved in the Serbia-Austria Hungary dispute of 1914. Even if this was not the case and Russia reacted exactly as per OTL, Britain would probably feel threatened enough to have some very serious qualms about joining the Entente in WWI...