If you mean the Manhattan Project during World War II, Germany didn't seem to know much of anything, and neither did Japan. Both countries knew about the possibilities of nuclear weapons but didn't think anyone else was significantly investing in them. If they had been looking more closely, they probably would have seen evidence of the work being done — the secrecy was far from perfect.

The Soviet Union knew much about the US and UK effort, because it had moles within the project. It knew quite a bit about the timetable of development, and of the specifics of the research. It knew that a test was being contemplated. Truman told Stalin they had a new weapon while at the Potsdam Conference, which Stalin correctly interpreted as an indication that the weapon had been tested successfully.

No country used technical intelligence to "spy" on the Manhattan Project — the means had not been developed yet. In fact, the US was just starting to develop such methods (e.g. checking river water for signs of radiation that would indicate it was being used to cool a nuclear reactor, or "sniffing" the air for radioactivity downwind of suspected sites) during the war, as an attempt to keep track of Germany's (stillborn) nuclear program.

On these matters, Jeffrey Richelson's Spying on the Bomb is a lengthy and authoritative source on all things relating to nuclear espionage and technical monitoring.