This is a very interesting topic and one which is probably unfamiliar to most of the readers. Because it existed as sovereign state and as a church, the Holy See has many characteristics of a country even though at the time, 1914, it was simply a part of Rome since territory was seized by the House of Savoy in 1870.
Pope Benedict XV would have had a great interest in the fate of the Russian Emperor as it would have had an effect on the Catholics of the Roman Rite living in Poland, Lithuania and Latvia; on the Catholics of the Ukrainian Rite living in Belarussia and Ukraine; and the Catholics of the Armenuan Rite in Armenia and scattered through out the Russian Empire (I know of Armenian Catholic Churches as far away as Tallinn and Saint Petersburg). At the time of the Russian Revolution, the official Catholic stance toward Socialism and Communism was very harsh, simply that it was incompatible with Catholic doctrine.
The Holy See of course would have been very interested in developments that would signify negative policies toward the Catholic flock by the leftist government. One would think that the Holy See would have anticipated the seizures of Catholic Church properties, convents, churches, monasteries, religious art, bank funds etc.
Through a large network of the religious and the observant, the Pope could direct that the nearest bishop send investigators to the Urals and report back on the fate of the IF.
The Revolution was very hard on the Catholic Church in the former Russian Empire, most of the buildings were confiscated, the religious art looted, bank accounts seized, bishops imprisoned and then in the 1920's the mass executions of Catholic nuns, priests, monks, and ordinary parishioners.
David