Germany could easily have nuclear weapons by 1956.
Cost is not that bad. The USA spent 1 billion or 2 billion USD, but did it 2.5 years. We built entire industrial towns that were then torn down. We did not have the theory worked out while doing the engineering. The Manhattan project has all the elements of a max cost project going grossly overbudget. Do this over time, you cut the cost at least by 75% IMO. Probably do plutonium bomb. So we are looking at about 250 million to 500 million USD spread over a decade or two. Compared to a pre-WW1 budget of about 600 million per year. It would be say a 25 million USD budget per year. Very doable. Less than the V-2 cost for example. Basic TL.
1) 1938 Fission discovered. Could easily be sooner, but we don't have TL details.
2) 1939-1943. Light funding, both medical military. Theory worked out.
3) 1944-1951. Seven year project. 25 million USD, under 10K workers. Test in 1951. Since we did slower rampup, we have nicely size reactors facilities to produce fuel for bombs. Making say 6 per year. By 1956 we have a nice size arsenal. Note: This is fairly low funding. It could be done a lot faster.
Germans aren't stupid, I'm pretty sure they'll understand the implications of creating and testing such a weapon, the same way that Leo Szilard and many scientists in OTL did. It is literally opening Pandora's Box, poorer, less stable nations will never undertake this project themselves unless they know it can be done. All the Germans will have achieved here is announcing to the world that it is possible to weaponise the nuclear chain reaction.