Looking at the various national grudges and assuming a similar spark for the world war in the Balkans, it could go down something like this:
The League of the Three Emperors is abandoned after Germany, Austria-Hungary and Russia fail to agree over Balkan policy. Contrary to OTL, Bismarck decides against sticking with AH in a Dual Alliance and instead prioritizes holding Russia at Germany's side in order to prevent a possible two-front war. So a strong German-Russian Alliance may persist for a while, which eventually leads to a lessening of war paranoia in and against Germany.
The Triple Alliance with Italy therefore never comes into being, and neither does the Franco-Russian Alliance and all the following military alliances. Instead, we would have a completely different set.
Possible war constellations over grudges:
1) Austria-Hungary against Serbia and Russia over influence in the Balkans
2) Italy joins in against Austria-Hungary over claims on Austrian territory in Trentino, etc.
3) Russia against Ottoman and British Empires over influence on Ottomans, and Middle East / Central Asia in general (i.e. Crimean War mkII)
4) British Empire against France over influence/control of colonies and imperial ambitions
5) Russia against Japanese over control of Manchuria (i.e. Russo-Japanese war mkII)
So we'd eventually end up with:
New Triple Alliance: British Empire, Austria-Hungary, Ottoman Empire, Japan
vs.
Franco-Russian Alliance & friends: Russia, France, Serbia, Italy
?