(This is a bit beyond the scope of the question, but I felt it was worthwhile regardless.)

Yes and no.

The US tried to limit access to information about nuclear weapons technology on the one hand. But on the other hand they were extremely permissive about the spread of peaceful nuclear technology.

The problem with the latter is that the systems used for nuclear reactors (solid fueled, low enrichment Uranium, light water/pressurized water reactors) are particularly proliferation prone, for several reasons. Moreover, the US and the west were generally uncritical of the spread of civil nuclear reactor technology without much concern for the very high potential for proliferation.

Weapons development relies on two cornerstones: acquisition of fissile material (weapons grade Plutonium or highly enriched Uranium) and the construction of a device capable of making use of fissile material to achieve a nuclear detonation. Of the two the second is by far the easiest to achieve, but that was the area that the US attempted to control. Constructing a nuclear bomb is a difficult endeavor even if you have the material for it, but it is something that is well within the engineering and technical capability of almost any advanced economy. Perhaps more importantly, it is something which can be done in near utter secrecy quite easily. Nations can spend years developing their own systems and testing them extensively with essentially no ability to monitor or impede such activities externally.

Now, when it comes to acquiring fissile material as I mentioned there are different kinds, Plutonium and Uranium, and each has their own difficulties. Enrichment of Uranium requires extremely sophisticated equipment (high speed centrifuges, gaseous diffusion plants, or calutrons) and the use of significant amounts of power. However, it was not fully appreciated until relatively recently how easily it is to build and operate such facilities in secret. Nevertheless, such equipment requires fairly advanced technical know-how and specialty knowledge of the systems, which the US did attempt to control, in some cases.

I should take a brief moment to talk about enrichment here. One of the traps that it's easy to fall into concerning enrichment is to look at the percentages. In that regard the enrichment level used in a typical civil reaction (say 3-5%) or even higher enrichment (say, 20%) may seem to be quite far distant from the level needed for nuclear bombs (80-90%). But what matters is the amount of work between different enrichment levels (the amount of "separative work units" necessary to reach a given enrichment level). Achieving 5% enrichment means doing 75% of the work toward producing HEU, and 20% enrichment is 90% of the work toward producing HEU. So furnishing a country with a significant amount of 5% enriched Uranium could potentially provide a huge leg up to a nation intent on acquiring nuclear weapons. If they wished to abrogate any agreements they've made and suddenly sprint toward a "breakout" nuclear weapons capability with materials on hand then being in possession of stockpiles of 5% or higher enrichment Uranium shortens any of the timescales toward producing weapon's grade HEU by a factor of 4 to 10.

But I feel I might be underselling the importance of this point. As I said the US tried to restrict the spread of Uranium enrichment technology, but only to a certain degree. There are nevertheless many nations who have been allowed to acquire or build significant Uranium enrichment capabilities. A few examples include: Mexico, Brazil, Japan, Argentina, South Korea, and Iran, among others. Such nations have the ability to build nuclear weapons if they so desire. As mentioned, enriching Uranium to sufficient levels for civil or research reactors is actually more difficult than taking low enrichment Uranium and producing weapon's grade materials.

Additionally, throughout much of the 20th century many countries were provided with research reactors, and often those research reactors were powered by small amounts of highly enriched Uranium.

When it comes to production of weapon's grade Plutonium the situation is even worse. Any nation possessing of fission reactors has the ability to subject natural Uranium (which is quite easy to obtain and quite plentiful across the Earth) to neutron flux and breed Plutonium. Indeed, all fission reactors naturally breed Plutonium. However, when the fuel is left for a long period in the reactor the Plutonium-239 is further bred into other isotopes of Plutonium, such as Pu-240, which render the Plutonium useless for bomb making. However, breeding weapon's grade Plutonium is merely a matter of removing fuel (or even special natural Uranium samples) from the reactor more often and then doing some fairly straightforward chemical reprocessing to separate out the Plutonium. This is protected against by various IAEA safeguards and monitoring, but again any country with even one nuclear reactor has the potential for a "breakout" capability to build nuclear weapons if they so desire.

If you look at the countries that acquired nuclear weapons in the 20th century (and who are not designated as nuclear powers by the NPT) you'll find that in every case they have been aided by civil nuclear research and engineering from either the US or an allied nation. For example, South Africa was aided by the US; Israel by the US and France; Pakistan by the US and Canada; India by the US and Canada. North Korea acquired nuclear weapons early in the 21st century largely through the aid of the AQ Khan proliferation network along with Soviet/Russian assistance (which also aided the Iranian program which currently possesses a latent or "breakout" bomb production capability).

In short, the US has put forth an effort to restrain nuclear weapons proliferation, but in reality much of its efforts have been either extremely naive or potentially disingenuous (i.e. using secretive means to slip nuclear weapons technology to favored states). My own opinion is that naivety is the better adjective for most of the US's efforts though in a few instances (particularly with Israel) there are some strong signs that there was a bit of back channel "deniable" active proliferation.

I would like to request clarification from experts to how the question is phrased.

OP referred to a period when only the US had nuclear weapons. My understanding was that this was not really ever the case, as the Soviets were already well along in their own development by the time the bomb fell on Hiroshima. But I know that RDS-1, their first test, didn't occur until 1949.

Is it reasonable to say that there was a period when only the US had nuclear weapons? Or was the simultaneous development in the Soviet Union effectively "having" them, even though they were a little behind?

I ask because it makes me wonder about modern nuclear powers. Does a country have to have actual bombs ready to go at a moment's notice to be a "nuclear power," or can they have the technology and process developed, but no actual bombs, to be considered nuclear capable?

The US took two approaches towards what we would now call non-proliferation, and what at the time was called "control of atomic energy":

• They passed a domestic law, the Atomic Energy Act of 1946, which sought to control how nuclear information and materials were contained within American borders. Among its provisions was to say that any information relating to nuclear technology had to be regulated (at penalty of death!) unless the Atomic Energy Commission declared it could do no harm, and no classified nuclear information could be given to foreign nations (any of them) until Congress declared that pretty much everything was peaceful in the world (e.g. never). (The Act was amended in the 1950s to permit some collaboration with allies.)

• They proposed a plan for the "international control of atomic energy" in the United Nations, known as the Baruch Plan, derived from what was called the Acheson-Lilienthal Report. It basically said that the United Nations would regulate nuclear technology worldwide and prohibit any country from making nuclear weapons, including the United States. The Soviets rejected it quickly, both because they were keenly interested in making nuclear weapons, but also because it required the USSR to submit to inspections and regulation before the USA would get rid of its existing stockpile, which they saw as both unfair and dangerous.

• Secretly, during World War II, they sought to buy up all known uranium and thorium ores in the world, as part of what was known as the Combined Development Trust. This was both from the point of view that the US program would use them, but also that they would be denied to others (like the USSR). This was why General Groves thought it would take the Soviets 20 years to get a bomb — he didn't think they had much uranium. (He was wrong, uranium is more common than they knew at the time.)

Other than that — they didn't do too much else in the brief period of the American monopoly (1945-1949).

Also how effective, if at all, were the steps?