I'm not sure I understand the question. If you mean prior to its invention, the idea of weapons that used atomic energy was around since the 1910s, but obviously they didn't know how something like that would work (they lacked a physical mechanism for releasing atomic energy on a massive scale). Once nuclear fission was discovered (late 1938, announced early 1939), the mechanism presented itself, in the form of a neutron-initiated nuclear fission chain reaction.

Early work stumbled on the difference between a reaction that used fast neutrons and one that used slow neutrons. This is essentially the difference between a reactor (slow neutrons) and a bomb (fast neutrons). So some of the early bomb ideas imagined that you would basically be making a reactor that would explode. If that is the case then you imagine these bombs to be gigantic — something that can fit on a barge — and not that powerful (you can't release enough energy fast enough that way).

The Manhattan Project started when a lot of people became convinced that you could do a fast neutron chain reaction with either enriched uranium (which is necessary for uranium to react with fast neutrons) or separated plutonium (a recently-discovered byproduct of nuclear reactor operation). Early on, the target explosive power was 10,000 tons of TNT, but at various times in the project the expected explosive power fluctuated a lot. They had weapon designs on the books that they thought would get less than 1,000 tons of TNT (<1 kiloton), but they weren't super happy about those.

These yield estimates are basically a function of the efficiency of the reaction — a question of how much material reacts before the bomb blows itself apart and stops the reaction. They were not especially confident that many of their early attempts would be very efficient. The earliest ideas of bomb design were variations on the gun-type design (make a critical mass by combining two sub-critical masses). There were many ways you could vary that kind of approach and they looked into a lot of them (e.g. "double-guns" that shot two pieces into each other simultaneously, thus doubling the final assembly velocity, but creating very difficult engineering challenges). They also looked into different types of implosion assemblies (collapsing material inward, or using high explosives to increase its density) as well. By the fall of 1944 they had decided that they would use a relatively slow-gun assembly for their enriched uranium, and a fast high-explosive implosion assembly for their plutonium. (There were other options on the table for both, but this decision made the gun problem essentially "solved" and allowed them to focus on what they hoped would be the most efficient implosion problem.)

By May 1945 they basically thought that the enriched-uranium gun-type bomb would have an explosive yield between 5-15 kilotons, and that the plutonium implosion bomb would range between 0.4 and 5 kilotons. You can see the lack of confidence in the implosion design in these figures. On the eve of the Trinity test, the most probable maximum yield was estimated to be 5 kilotons. The test ended up being around 20 kilotons, i.e. it was 4X more efficient than they had estimated.

As for ignition of the atmosphere — they had wondered about this in mid-1942 but had quickly shown that even with very radical guesses about how easy it might be to start a fusion reaction between the nitrogen atoms, that it wouldn't propagate. This idea of course stuck in the back of some of their minds (after all, they were trusting a lot to calculation), but from a scientific standpoint nobody believed it would happen. Fusion is hard to pull off even on a limited scale.

All in all I think one can say that those making the bombs under-estimated how powerful they would be. They also under-estimated their mortality: Oppenheimer thought that the Hiroshima bombing would only kill around 20,000 people, it ended up being at least three or four times that figure.

As for the general public — in general the US public had an under-estimation of the powers of nuclear weaponry in the early nuclear age (e.g. the 1940s), believing they were more easily survivable than was probably true. This was in part a misunderstanding of civil defense instructions, which never claimed everyone would survive, but rather than certain activities (e.g. bomb shelters, duck and cover, etc.) would decrease overall mortality and injury rates.

When thermonuclear weapons came around (the 1950s), people started worrying much more about the size of the bombs, and the potential of radioactive contamination. This is where you start to see the understanding go in the other direction, in terms of exaggerated effects — one bomb and the whole world is one fire, etc. Which is not quite true, either, though multi-megaton weapons are indeed pretty powerful (destroyers of metro areas).

I'm not sure if this answers your question, because I'm not entirely sure who "the people" are in your question. My favorite book on attitudes towards nuclear weapons and nuclear energy over time is Spencer Weart's Nuclear fear: A history of images (1988) which is both scholarly and readable.