So let us assume that either Moscow or Sankt Petersburg would have been destroyed. First of all Russia would have been weakened, and it would have reduced its aspirations at the Balkans. This would have reduced the tensions between Russia and Austria-Hungary. But this would not have made the situation on the Balkans more peaceful, as every ethnic group there had some issues to settle with its neigbours. Indeed, there were two Balkan wars in 1912/13, without (direct) participation of russia. So in this situation Austria-Hungary might have tried to get more territorial gains. This might have made this already boiling ethnic mixture of austria-hungary more fragile, with everything possible up to a civil war.
But let us assume that a whole-european war would have been delayed, just because of the critical role russia played at the actual outbreak of WW1. in germany two things would have happend that didn't happen due to the WW1. The german fleet programm, which enraged the british empire, was already seen as a failure just before the world war, as it neither pressed the united kingdom in a alliance nor would be able to protect germanys trading routes. in the best case, germany would have reduced its fleet program and this conflict with the united kingdom would have ceased over time. in the worst case, germany would have tried to adapt its fleet programm and thus the conflict would have remaind. In the domestic politics conflicts between reforming forces (among them the social democrats which just got 40%+ seats in the last election before the war) and the militaristic conservatives threatened to paralize the german empire. With the grat war being delayed just for some years, reforms of the very restrictive constitution of the german empire would have been ineviatable. Under certain circumstances, this might have lead to a constitution were germany would have been ruled more by the people and less by the military and the emperor. This then might lead to a far less agressive tone toward the other great powers. But the so called hereditary enemyship between france and germany would have still remaind, as this was still in the heads of the people of both nations.
In this alternate history it would be hard to predict who would lead the next war. most certainly france and the german empire (even a more "liberal" empire) would sooner or later clash. Britain would surely join france, because even if the direct enemyship between germany and the british empire would have ceased, the prospect of having a german hegemony over continental europe would not be acceptabel by the Britains. Austria-Hungary (if it still excistet...) would join germany, as many of its people were german by culture and still mentally connectet to the german empire (the feelings of the non-german people about this would be a major problem for austria-hungary). Russias actions would depend on how the Sankt Petersburg/Moscow event would have affectet the empire. Likely it would still try to rebuild and ignore the war, perhaps it would join the war but remain weaker than it was in the actual WW1. Perhaps the russian empire would be in a state of cofusion or chaos and ignore the war, but this would mean a complete redirection of the foregin policys of all the other empires, making my speculations completely pointless
. The USA would not join the war at all, as i think a less militaristic germany would not attack us civilian ships (and even during WW1 it took three years to engage the war). Italy and the Ottoman Empire are the hardest to predict. The ottoman empire would most certainly not join the allied side if the russians were part of it. In WW1 its parlament very very barely deciedet to side with the germans, so the ottomans would either remain neutral or perhaps join the german side. Italy might join if the allied were the aggressors, otherwise it might behave pretty much as it did during WW1, not joining germany and joining the allied if the other powers would offer something.
