Edit: oh, well, just noticed that my PoD falls one year just short of OP guidelines. Sorry. I however suppose that with some adjustment, the general idea of Aistria, Italy, Russia, and Turkey joining the fun in 1870-71 may still yield the desired scenario nicely.
Besides the Berlin Congress PoD which Onkel so masterfully developed, there is another handy PoD (a variant of
this TL) for this alliance, that is an expansion of the the Franco-Prussian War to Austria, Italy, and Russia, with this conflict and the Russo-Turkish War getting wrapped into one.
It goes this way:
Some changes in the Italian high command ensure that Italy reaps decisive victories in the battles of Custoza and Lissa. Faced with a two-front defeat, Austria is forced to ask for a beggar's peace. Italy asks for all of its irredentist claims (Venetia, Trentino, Istria, Dalmatia), which prompts the Prussian King and generals to expand Prussian war claims and include Saxony and Bohemia-Moravia, overruling Bismarck. Alarmed at the size of Prussian-Italian success, Napoleon III switches to a pro-Austrian diplomatic stance and threatens an intervention if the demands of Berlin and Florence are not scaled down. Unwilling to risk a two-front war, Prussia and Italy comply with French demands. Final peace treaty gives Trentino (not South Tyrol) and Gorizia-Gradisca to Italy, Saxony and German-majority districts of Bohemia-Moravia (modern Sudetenland) to Prussia, in addition to OTL gains.
Austria is thrown by the defeat into severe instability, but the timely reform of the empire to an Austro-Hungarian real union, with the grant of autonomy to Czechia and Croatia within the two subunits, and the union of Croatia and Dalmatia, stabilize the Danubian monarchy somewhat. Tensions between France and the Prussian-Italian alliance grow and Austria, eager to secure protection, makes an alliance with France. Bismarck counters this by making a secret pact with Russia, which guarantees Russian support against Austria in exchange for Prussian support to Russian expansion in the Balkans.
Within a few years, France declares war to Prussia and Italy about any or all of several possible casus belli (Luxemburg, Rome, Spanish succession). French aggression unleashes a wave of German nationalist sentiment in the southern German states, which side with Prussia. It also happens in Austria, but Franz Joseph is able to quash Pan-German agitation with the support of the Magyars, Czech, and Croats, so Austria joins the war. True to its pact with Prussia, Russia intervenes.
The war leads to a defeat of France and Austria, and the formation of the German Empire. True to their word, Germany and Italy support Russian expansion in the Balkans, and Russia, soon joined by Greece, Serbia, and Romania, expands the conflict to the Ottoman Empire, with a defeat of the latter. Alarmed by Russian expansion, Britain threatens an intervention, and the German-Russian-Italian alliance, militarly overextended and financially exhausted, accepts a compromise peace.
In the following peace conference, Germany annexes Alsace-Lorraine, Bohemia-Moravia, and Tyrol, Italy annexes Nice, Savoy, Corsica, and Istria, Russia annexes Galicia and some Ottoman Georgian and Armenian territories in the Caucasus. Serbia and Romania become independent (and Russian satellites), while Greece gains Thessal. Bulgaria and Bosnia become autonomous tributary principate under Ottoman suzerainty with a Christian government. Britain gets administration of Cyprus, while Tunisia and Libya are recognized in the Italian sphere of influence and Morocco in the German one.
In France, after some serious political convulsions caused by the defeat (fall of the Napoleonic regime, far-left insurrections in Paris, Lyon, and Marseilles), a reactionary rmonarchist regime takes over in France and the Bourbon dynasty is restored on the throne. Conveniently enough, the heir to the French throne is also the Carlist pretendent to the Spanish throne. The Carlists win the Third Carlist War with French support, and a Franco-Spanish real union is established.
Soon after the peace conference, the German-Russian-Italian alliance is confirmed as the League of Three Monarchs, later redubbed the Triple Alliance, informally joined by Serbia, Romania, and Greece. France-Spain, Austria-Hungary, and the Ottoman Empire confirm their own alliance too, which gets later dubbed the Triple Entente, informally joined by Bosnia and Bulgaria. Britain remains a pro-Entente neutral for a couple decades, then joins the TE out of growing alarm about the power of the TA continental block.