Henry Wallace becomes president but is poorly prepared for the job, especially without an experienced vice president to delegate to. The war ends up taking longer due to poor delegation, poor relations with Congress, and general dysfunction. Secretary of State
Cordell Hull would be the next in line for the presidency and would be qualified to do so due to his long experience at the State Department, but Hull was in poor health and ended up resigning in late 1944. It's quite possible to see World War II lasting longer at greater human and material cost for the United States, its allies, and the people of Asia.
With all the dysfunction in the United States, the United Kingdom might decide to work on
Tube Alloys instead of working with the United States on the Manhattan Project. The United Kingdom played a major role in helping to move the Manhattan Project along, including by supplying the United States with uranium from Canada and the Belgian Congo. If the United Kingdom decides to keep those resources for itself that well significantly set back the United States effort, and it would result in the early nuclear arms race being between the United Kingdom and the Soviet Union.
The United States would likely elect a more isolationist government in 1946 or 1948 (both the Wallace Democrats and traditional Republicans leaned towards isolationism), allowing the European powers to play more of a role in global affairs. However, the United Kingdom and France would both be in a weakened state, and if the Marshall Plan fails to pass it would leave them even more weak. The Soviet Union would begin to surpass them in influence and both Italy and Greece could come under communist control, just like Czechoslovakia. Winston Churchill and other hardline anti-communists would play a prominent role in the United Kingdom, and something like
Operation Unthinkable could play out in the 1950s.
Eventually the Soviet Union would start to pull ahead of the United Kingdom in the nuclear arms race due to its greater resources. The United States could lose years of time working out the issues of nuclear control and ownership, issues that historically took until the mid-1950s to fully resolve. Historically the United States took the unprecedented step of creating the
Joint Committee on Atomic Energy, but if something like that doesn't exist there won't be the major structural advantages that helped the development of nuclear technology in the United States, especially for civilian purposes.
In our timeline the
Atomic Energy Commission spent its first decade far more interested in military projects than civilian ones, viewing nuclear power as something that wouldn't be worth seriously considering until around 1980. That created a unique environment that allowed the United Kingdom and Soviet Union to become early leaders in nuclear power. In fact, the first commercial scale nuclear reactors were built at
Calder Hall in the United Kingdom, and the British went on to export
Magnox units to Italy and Japan for their first nuclear power plants.
The opening of Calder Hall was a big wake up call for the United States that led to
Atoms for Peace and programs to help develop private nuclear power in the United States. The first commercial scale nuclear power plant in the United States,
Shippingport, was actually a design that was originally intended as a propulsion unit for a never built predecessor to USS
Enterprise. It used highly enriched uranium and was totally impractical as anything other than a demonstration plant, but the United States might not even have naval propulsion reactors or large amounts of highly enriched uranium that are available if it is behind in its nuclear program.
The scenario that first led the United States to develop nuclear reactor technology could play out, with the United Kingdom, Soviet Union, and to some extents France all offering access to it as a reason to join their spheres of influence. More countries might choose to take up offers of nuclear assistance from the Soviet Union if the United Kingdom and France are unwilling or unable to offer anything similar, especially if the United States isn't in a position to offer an alternative. There will also be more diversity in the types of nuclear reactors without the influence of naval propulsion, perhaps leading to the popularization of designs using graphite or heavy water moderation for civilian purposes instead of instead of light water. That would get around the need to enrich uranium fuel, allowing nuclear power to be done with less need for imports and/or enrichment infrastructure.
The world was heading towards having dozens of nuclear armed states prior to the Nuclear Non-Proliferation Treaty being established in 1970, and even then it took 25 years to become permanent and nearly universal. Even if nuclear weapons aren't used, the international situation will likely be one of individualism that leads countries towards acquiring nuclear weapons of their own. It wouldn't be too difficult to acquire nuclear weapons from a research reactor or a
Magnox/
UNGG type reactor, especially with the lax safeguard standards that were common prior to the
1974 Indian nuclear test. As the number of nuclear weapon states increases it only further increases the chances of them being used in an international or civil conflict, with each incident only further fueling proliferation.
The resulting world would be a multipolar one having more in common with the days prior to World War II. It would also be one with more nuclear weapons and more states having nuclear weapons. Authoritarian governments could use them as a shield to prevent foreign interference and even domestic conflict. If things became destabilized, perhaps non-state actors might even acquire some nuclear weapons of their own.