As far as we know, the answer is that the Axis countries did NOT get much information on the Manhattan Project. Part of this is that Axis countries had notoriously crap intelligence when it came to the United States — they never made significant penetration. The other aspect appears to be that while there was some relevant information buzzing around in non-US contexts, the Axis intelligence agencies never seem to have put it together into one coherent picture. In a sense, this isn't too surprising: the project was indeed HUGE but it was also spread out and heavily compartmentalized. Most of the people who worked on the project didn't know what they were doing or where the other sites were.

If the Germans had thought to really look for the Manhattan Project, they probably would have seen signs of it. There were indeed many leaks. But they were each piecemeal and incomplete.

I've discussed what little the Nazis seem to have known here. There are indications that they got whiff of rumors and small leaks. But they don't appear to have taken them very seriously and they don't seem to have realized what they meant. Given that their own scientists were telling them it was essentially impossible to pull off such a thing, it is not surprising they didn't push more.