This doesn't happen quite the way Jackson described it1. When Elrond tells his story to the Council, the moment is described as more like a group discussion, rather than Elrond literally dragging Isildur up to Sammath Naur (emphasis mine):
'Isildur took [the Ring], as should not have been. It should have been cast then into Orodruin's fire nigh at hand where it was made. But few marked what Isildur did. He alone stood by his father in that last mortal contest; and by Gil-galad only Círdan stood, and I. But Isildur would not listen to our counsel.
'"This I will have as weregild for my father, and my brother," he said; and therefore whether we would or no, he took it to treasure it.
Fellowship of the Ring Book II Chapter 2: "The Council of Elrond"
So there was somebody else present: poor old Círdan, but that was it. As suggested by the question, Isildur claiming the One Ring happened very quickly and quietly, and there was just nobody else near him at the time.
The above also seems like a reasonable explanation here. Consider Elrond's tale:
It seems clear that Elrond ran up to Isildur within moments of Isildur picking up the Ring; the poor guy hasn't even had the chance to stand up yet. Unfortunately, we don't have a confirmed reason why Elrond didn't bother to grab a couple of Elf-soldiers on his way to reach Isildur. However, there are some possibilities:
If Elrond knew of the corruptive effects of the Ring at the time (admittedly a question that we can't definitively answer), he probably hoped that if he could just get Isildur to Sammath Naur fast enough, Isildur would be able to destroy the Ring before it grabbed hold of his mind. He probably didn't have time (or didn't think he had time) to grab another Elf or two, give them an order, and then have them follow him all the way up Mount Doom
The other Elves may not have been recovered yet. In the prologue, we see that Sauron's defeat causes a massive shockwave that levels all of the forces (including the Elves) in the area. We don't see this to be sure, but it's possible that there weren't any Elves around Elrond who had gotten up yet and, related to the above point, he may not have felt like there was time to wait for them
We don't know if the battle was still going on. This is admittedly a shakier idea, considering the orcs stop fighting pretty much immediately when Sauron is destroyed in Return of the King, but it's possible that the rest of the Elvish army was still working on killing (or chasing) what was left of Sauron's army. Knowing the orcs, the ones who were left were probably running as fast as they could away from the fighting
1 I seem to say that a lot when answering questions based on the movies. It's almost as if movies and books are different mediums, which require different storytelling techniques...