1) The A-4 reached 176 km into space. It did not have a warhead during that test. (or even a dummy warhead) It was lightened. Launching a satellite is lots of technical issues beyond just a rocket that goes to height X. It's like building a tank is a lot more complex than just putting a cannon on a truck with some armor.
2) I'm not saying the US was the only one who could pull it off. I'm saying the US did and it took a lot of people working in a lot of areas, undisturbed by the war, to do it.
Could the Germans have done it? Possibly, as you said, we don't know the exact size, but historically it was hundreds of people working on it in Germany. Peenemunde had 5000 people at it's height in 1942 and that was with the full and enthusiastic backing of the Reich. That's probably what the Germans could have put into Nukes. It's not enough. They would have also had to put all of the facilities underground which raises the cost and they would have needed to pull a lot of people from more promising projects.
I do know that it took subsequent nuclear powers about a decade to develop nukes start to finish. Those were all modern states with more understanding of atomic weapons and more resources than the Germans had in the early 1940's. The Germans were doing interesting research and had isolated radioactive isotopes. They hadn't figured out how to make an actual bomb that would undergo fission.
3) Building a nuclear reactor was a significant step, but not the only one.
4) Ok, my point with the Maus was that the Germans had a lot of great ideas that were impractical and needed a lot more testing when they were actually built. They built a bomb of some sort with fissile material but they didn't achieve a nuclear detonation because they hadn't got the math right (or something, we don't know.) In any case, they couldn't test as fast as the Americans (the graphite problem) and they couldn't calculate as fast either. (The US put hundreds of mathematically inclined young men into small rooms and made them work as "human computers" doing all the mathematics that the physicists gave them. The Reich didn't have that many people to spare)
The German nuclear program would have been eventually successful with full funding. But that eventually is definitely a ways beyond 1945. They lacked too many things to speed up the process even if they wanted to.