Uranium was first purified in 1841. Radioactivity was discovered in 1896 when Becquerel left a uranium sample out near an undeveloped photographic plate and followed up on the discovery that the plate was clouded. This was an accident that could have happened at any time after the purification of uranium, so let's move it up to 1848.
IOTL, neutrons, fission, and isotopes (the scientific prerequisites for attempting a fission bomb project) were discovered over the course of about 40 years from the discovery of radioactivity. TTL, there would be fewer resources available for research and less scientific groundwork in other areas, so we can expect research into radioactivity to go significantly slower without a big boost from somewhere else.
Enter the reason for my choice of 1848 as a POD -- the Revolutions of 1848. Europe in general and Germany in particular has seen a wave of attempted revolutions, King Fredrick William IV of Prussia has declined an Imperial throne offered "from the gutter" of the revolutions and has instead imposed a monarchist constitution on Prussia, and Otto von Bismark has been elected to the new Prussian Landstag established by FWIV's constitution (his first elected office) as a representative from Saxony.
Meanwhile, a chemistry student at the University of Leipzig had decided to reproduce some experiments he'd read about to purify uranium. He accidentally irradiates an undeveloped photographic plate, notes the significance, and brings his discovery to the attention of two of the physics professors at the university: Wilhelm Weber and Carl Gauss, who perform their own follow-on experiments and realize the vast potential of this discovery. Gauss contacts a few local Landstag representatives about seeking government funding for the research, and Bismark in particular is intrigued by the possibility of channelling the rising feelings of German nationalism into pride in scientific accomplishment, and becomes a major sponsor of funding for research into radioactivity throughout his career.
Optimistically (in terms of pace of scientific advancement), TTL may realize the possibility of a fission weapon by the mid-1880s (35-40 years after TTL discovery of radioactivity), triggering a race between the Great Powers to develop such a weapon. Germany and Britain conduct successful nuclear tests within months of each other in the early 1890s, and both develop Teller-Ulman style H-Bombs by 1900.
Now, nukes are the hard part of a Project Orion-style space race. The vehicle itself is a significant engineering feat that may require a decade or more of well-funded work to fully develop, but the prerequistes for this work is roughly the technology needed to build a Dreadnought battleship. We have this know-how as well as the nukes by 1900 ITTL, as well as a cold-war situation between Britain and Germany to motivate a space race, leaving us 14 years before the 66-year window post-POD for this space race to land a ship on the moon.
