In survivor accounts from Hiroshima, many of them initially report that they thought it was a conventional bomb that just happened to hit their house. They also then gradually realize that the damage was universal, and that it must have just been one incredibly large bomb. E.g. from the account of Father John Seimes, a Jesuit who was at Hiroshima:

I realize now that a bomb has burst and I am under the impression that it exploded directly over our house or in the immediate vicinity. ... We proceed to the front of the house to see where the bomb has landed. There is no evidence, however, of a bomb crater; but the southeast section of the house is very severely damaged. ... Down in the valley, perhaps one kilometer toward the city from us, several peasant homes are on fire and the woods on the opposite side of the valley are aflame. A few of us go over to help control the flames. While we are attempting to put things in order, a storm comes up and it begins to rain. Over the city, clouds of smoke are rising and I hear a few slight explosions. I come to the conclusion that an incendiary bomb with an especially strong explosive action has gone off down in the valley. ... [Another priest in a different location] thought that the bomb had fallen in his immediate vicinity [too]... Soon comes news that the entire city has been destroyed by the explosion and that it is on fire. What became of Father Superior and the three other Fathers who were at the center of the city at the Central Mission and Parish House? We had up to this time not given them a thought because we did not believe that the effects of the bomb encompassed the entire city.

Etc. One thing that Seimes repeats quite a lot is what whenever they meet someone else, they think a bomb had exploded right on top of them. You can see how scattered sensations gradually led to an understanding that there had only been one bomb. They did not know it was an atomic bomb until after it had been announced as such by the United States. Even then it took some time for the Japanese people get a full sense of things (it was not until just before the Nagasaki bombing that Japanese newspapers carried more or less accurate stories about the atomic bombing of Hiroshima).

As an aside, the local effects were quite different in many ways from conventional bombing. For one thing, conventional bombing raids used so many planes that you could hear them coming, and did not take place instantaneously. They were slower attacks, and you could flee them. The atomic bombs were more or less instant with their wrath, and their blast effect was much more important for destruction than it would be in a conventional or incendiary bombing raid. (As with incendiary bombing, there was a firestorm at Hiroshima, but it was formed by secondary fires.)

The way they first saw that it was a nuclear bomb was that in the hospital, doctors saw that all of the x-ray film was fogged, despite not having been exposed yet.