In very brief:

• US did it with help from scientists in the UK, Canada, and many European refugees from a number of countries (France, Poland, Denmark, Italy, Germany, etc.). But not Nazis or ex-Nazis. (The US missile program is a different story on that front.) When you think "Germans who helped the Americans get the bomb," you should be thinking about German refugees from the Nazis (some Jewish, some not), not Nazis.

• The Germans were nowhere close to making a bomb, to the contrary of many a thrilling "race for the bomb" narrative. They had a modest nuclear reactor research program.

• The Soviet Union started working its nuclear program in a minor way during WWII, and in a major way after it. They tested their first bomb in 1949. They benefitted in part from former German expertise, and from espionage information gotten from spies working for the US and British programs.

• The British decided to build a bomb in 1947, after they were cut off from any further input into the US nuclear program following the passing of the Atomic Energy Act. They took advantage of the aforementioned refugee labor (not ex-Nazis), as well as their experience on the US program. They set off their first weapon in 1952.

• The French started a bomb program in the 1950s, in a bid for relevance and independence. They had some people who had worked on the Canadian part of the American program (complicated, eh?). They detonated their first bomb in 1960.

• The Chinese started their program in the late 1950s, after being threatened by the US explicitly with "nuclear blackmail" during the Taiwan Strait Crisis. They received initial help from the USSR, but after the Sino-Soviet split they had to re-learn a lot of the work, because the USSR pulled its technicians out and didn't tell the Chinese how any of it worked. The Chinese detonated their first weapon in 1964.

• Israel fits in here somewhere, though it is not clear they ever detonated a weapon (they might have in 1979, but it's a little mysterious). They received help from the French in this endeavor. They were nuclear-capable by 1967 though their initial capability was quite crude.

• After a war with China, India began to vigorously pursue making a weapon in the late 1960s. They detonated a "peaceful nuclear device" in 1974. They did not benefit from deliberate help, but they did manage to use Canadian and French peaceful nuclear technology to their advantage. They detonated a "weaponized" nuke in 1998.

• After India developed a weapon, Pakistan pledged to get one too. They had a very long, secret program. They did steal some centrifuge plans from the Netherlands and used them, and received some help from the Chinese. They were nuclear-capable by the late 1980s, but only tested a weapon in 1998.

• South Africa also secretly developed nuclear weapons, with some help from Israel. They had about six crude weapons by the late 1980s but never tested them (so we think). They disassembled them in the early 1990s and renounced nuclear weapons capability, signing the Nuclear Non-Proliferation treaty (NPT).

• North Korea pursued nuclear technology on and off, before finally withdrawing from the NPT in the early 2000s and building and then detonating a weapon. They had benefitted from early Soviet assistance, and much later from purchased Pakistani technology and knowledge.

So that's an overview. There are many more wrinkles to every case. One common theme is that no nation really did it entirely "on their own," but none of them were really "given" the technology. Even in the cases of direct technology transfer (e.g. the French helping the Israelis build a reactor) there was still a lot of "independent" work to do.

There are many reference works on this subject. For a broad overview, Jeffrey Richelson's Spying on the Bomb is a succinct history of proliferation efforts, as seen from a US intelligence point of view (and includes unsuccessful efforts, like the Taiwanese bomb project).