I think the the British, the Germans and the Japanese could prove to be the powers more likely to obtain nuclear weapons, but it won't be easy.
The British were the first ones to discover nuclear fission and in March 1940, discovered that as little as 1 kilogram of pure uranium-235 would be enough to create a nuclear explosion. The Manhattan Project could not have gotten off the ground without British help. The problem is that the British do not have the manpower or the resources of the United States. Without WWI, I suppose they could've went to their colonies for resources and manpower, but without WWI, the fate of Europe's colonies is up in the air.
There is also a bigger problem that would inhibit a British nuclear weapons program: the economic burden.
When there was a serious gap in the British-American scientific relationship during World War II, the British crunched the numbers on how much it would cost to build their own atomic bomb, without outside assistance.
3,000,000 pounds in research and development
5,000,000 pounds to build a nuclear reactor to create 1 kilogram of plutonium
Between 5,000,000 and 10,000,000 pounds to create facilities to produce heavy water
500,000 tons of steel
5000,000 kilowatts of electricity
If the British wanted to create their own nuclear weapons program in this timeline, it would be a burdensome effort and I highly doubt you would see a government crazy enough to authorize such a program, unless there was a world war.
Before World War I, Germany was effectively the heart of the European scientific community. It's where you went if you wanted to study something serious. If anti-Semitism can be suppressed, then I think the Germans could use their reservoirs of Jewish scientists to get a head start on the nuclear program.
As for uranium, in OTL, the Soviets and East Germans effectively scarred the Ore Mountains by mining it for uranium and during the Cold War, East Germany was the world's third biggest producer of uranium ore behind the US and Canada, so Germany definitely has the uranium resources to create nuclear weapons.
As for heavy water, I suppose Germany could lean on Norway for assistance in that regard. In OTL, Norway did have the Norsk hydroelectric plant. I don't think Germany could've produced heavy water on it's own, but I could be wrong.
As for Japan, they have the brains to get a project going, as even the Manhattan Project Intelligence Group admitted. In the 1930s and '40s, the Committee of the Application of Nuclear Physics was led by famous Japanese physicist Yoshi Nashida.
During World War II, the most successful Japanese nuclear weapons project was the Japanese Navy's F-Go Project, based in Kyoto, which by September 1945, had obtained 20 grams a month of heavy water from ammonia plants in Korea and Burma. Speaking of ammonia, in 1926, industrialist Jun Nuguchi had founded the Korean Hydroelectric Company in what is now Hungman, North Korea. This site eventually became the site for fertiliser production and contained a heavy water production facility whose output rivalled the Norsk Hydroelectric Plant in Norway. Japan never used this facility in their efforts to produce nuclear weapons, however. Have a Japanese nuclear weapons project use this facility and they'd have limitless heavy water.
The problem, however, was that the Japanese nuclear weapons project, of course, needed uranium. In 1945, the Japanese had requested the Germans ship them 560 kilograms of unprocessed uranium oxide by U-Boat. This U-Boat surrendered to US forces in April 1945 after Germany's surrender. So, Japan would either be dependent on foreign powers for the uranium (which I doubt they were just going to hand over) or you would have to have the Japanese extensively explore regions such as Fukushima Prefecture in the Home Islands and Korea, just as they did in our timeline, with the express intent of acquiring uranium.
So two out of these three nations would have to rely on outside assistance if they want to get anywhere close to building a nuclear bomb if they lack the resources in their own country or if the cost is simply too burdensome and depending on what prevents WWI in this timeline, I doubt you would see much interest in nuclear weapons.