I've written as some length about the timeline of events, and why historians are largely of the opinion that the Nagasaki bombing did not play a large role in the decision for the Japanese high command to surrender. That does not answer the question of "necessity" which is more than that.

Here is how Barton Bernstein puts it:

This bomb [Nagasaki] was almost definitely unnecessary. Without that bombing, Japan would have surrendered-very probably on the 10th. Amid the continuing blockade and the heavy conventional bombing, the powerful hammer-like blows of the Hiroshima weapon on the 6th and Soviet entry on the 8th had emboldened the peace forces in the Japanese government to try to end the war with only the single condition of the guarantee of the emperor. The day before the Nagasaki bombing, the emperor had agreed to intervene in deliberations, if necessary, to end the war with that one condition. “We must put an end to the war as speedily as possible,” he had instructed the Lord Privy Seal Koichi Kido.” Undoubtedly, Hirohito would have intervened without the Nagasaki bombing to plead, successfully, for Japan’s surrender.

Barton J. Bernstein, "Understanding the Atomic Bomb and the Japanese Surrender: Missed Opportunities, Little-Known Near Disasters, and Modern Memory, " Diplomatic History 19, no. 2 (Spring 1995), 227-273.

In another place, I believe Bernstein phrased it along the lines of: we can see that in the fateful Cabinet meeting of August 9th (which was instigated by the Soviet invasion, and during which the Nagasaki bomb was reported), the Nagasaki bombing news did not change anyone's opinions from what they entered it with. I think this is a fair judgment.

One can imagine some kind of more subtle impact, as people often do (e.g. "now they knew there were more than one bomb"). But if it did have an impact, it was very subtle indeed, to the point of being impossible to pick out of the other influences, which is to me essentially the same point that Bernstein is making.