I'm going to handwave something and imagine that someone makes a theoretical breakthrough on an Einsteinian level and the idea of nuclear fission becomes known in 1905.
What now? Well, you need to convince a country (for at this point in history, only a nation has the resources needed) that a bomb is possible and that one should be developed. Out of all the Great Powers, Germany has the need for an equalizing force against Great Britain, France, and Russia. It is superior to the rest in chemical and materials technology. It has a capable education system that can produce the engineers, physicists, chemists, and other scientific specialties needed. It has the ability to spend vast amounts of money on a single project thanks to its relatively authoritarian structure. That's the good news.
But you have massive difficulties involved. In 1905, scientists haven't even discovered the idea of isotopes, for Christ's sake! You'll need massive investment to even make a bomb possible on a technical level before you can even begin work on the tools to build the bomb itself. You don't even have the tools to identify the problem, let alone the tools to build the tools to get the uranium or plutonium to build a bomb.
And uranium enrichment -- forget about it. Even if scientists knew that there were different isotopes in 1905, there's no way to reach the uranium purity needed. Gaseous diffusion won't work because you don't have the special filters needed to remove the heavier isotope. Electromagnetic diffusion can't reach the quality needed. Centrifugal diffusion is technically possible, but laser diffusion, cryogenic diffusion and chemical diffusion are all out.
In the end, your best choice would be to focus on plutonium enrichment. But the cost, my God! To jump generations of technology, the cost goes up exponentially unless you're willing to spend decades on the project. In OTL, the Manhattan Project was by far the single largest engineering project of the war, roughly equivalent to the construction of America's entire fleet of strategic bombers. Imagine what it would cost in 1905, even ignoring uranium bomb enrichment! It'd be like building the Great Wall of China.
Given the resources available, it's utterly impossible. You're simply starting from too low a base and there are too many questions to answer. Let's say you somehow got William II to buy into this insane "bomb" idea. He'd have to cancel the German dreadnought program and pour every bit of money that was spent in OTL on those ships into this program and more billions besides. It would be a national undertaking, impossible to keep secret. Even if William somehow managed to keep a firm hand on the project, the country would fall to pieces under the burden.