As the title says, the entire reason for the cold war being cold was the threat of MAD. But how did the Soviets acquire the technology to develop their own nukes? I assume it would've been a very closely guarded secret by the US, and developing them from scratch would take decades. Were multiple countries, including the soviets, working on nuclear weapons programs even before Hiroshima and Nagasaki? Was the theory already well known and only some details needed to be worked out by the Manhattan project? Or did the soviet espionage acquire so much info that they were able to jump start their own nuclear missile production?
The Soviets had started a modest nuclear program prior to Hiroshima and Nagasaki, tipped off to the US-UK program by their own spies. They did not have the resources during World War II to mount an all-out crash program, but they acquired information and materials and began the groundwork. After Hiroshima, Stalin ordered a bomb to be made as soon as possible, and huge resources were allocated to this task under the direction of Lavrenty Beria, the feared head of the NKVD, which not only was the Soviet "secret police" and spy agency, but also ran the GULAG system that provided the slave labor for uranium mining, plant construction, and so on.
It was a massive effort. Yes, they had access to some espionage, esp. about the design of the weapon used at Nagasaki. But they did not simply "copy" in a rote way; they used that information as a guide and a "check" on their own scientists (because Beria didn't trust them, either, and he didn't trust the espionage fully). They also had some German and Austrian scientists whom they looted from Nazi Germany who helped them with specific parts of the work (but whom they also didn't totally trust).
The bulk of the project involved locating and mining huge amounts of low-grade uranium ores and processing them in the way that one must to build industrial-sized nuclear reactors. They then used these to make plutonium and then design and test a bomb by 1949. The Manhattan Project took about 2.5 years to get a bomb, the Soviet program took about 4 years to get one. So it was fairly speedy — a true crash program — but it was not instantaneous. Note that the "consensus view" among US scientists in 1945 was that the USSR could have a bomb within 5 years if they wanted to; the reality was not so far off of that, and the "consensus view" was not really developed with any firm methodology or anything.
There is and was no "secret" to the atomic bomb in the sense of one bit of information that either gives you it or doesn't. There are many "secrets" — things the US tried to keep secret, anyway — some of which were compromised by spies, some of which the Soviets independently figured out because these "secrets" are really derived from facts of nature that any sufficiently funded and talented scientists could find out. There was also significant information about the underlying processes (like nuclear fission) published before the war, and the US also published a lot of "high level" information about the size of its plants, which methods worked, etc., after the war. So the Soviets had a pretty good idea of how the US had done their work, and modeled their own project on the US program, because they knew the US had done it quickly and effectively (and Beria didn't want to fail!).
The Soviets had entirely capable scientists at their disposal and huge budgets and huge amounts of (GULAG) labor. Building an atomic bomb is not about coming up with the right equation (though you do need some theoretical work), it is about building an entire nuclear industry from scratch. Building industries from scratch is something that the Stalin-era USSR was actually pretty good at, having essentially industrialized the country very rapidly already in the 1930s. I would not put too much emphasis on the espionage for their success; it is clear the Soviet scientists were clever enough to design bombs on their own (and indeed, did come up with better bomb designs than the 1945 bomb design the US had, but Beria wanted the first one to be a "sure thing" so kept them to the old design), and they did have to build the industry essentially on their own (the spies couldn't really help them there; the spies did not have detailed industrial information and "know how" about how to, say, manufacture diffusion barriers or even reactor fuel). If the espionage saved them any time, it was a small amount of time — months, maybe. It's likely the espionage didn't save them any time, because "knowledge" was not the part that slowed the Soviets down in any real way — it was their lack of raw materials (they had very poor uranium resources).
For a lot of detail into the how, David Holloway's Stalin and the Bomb is a classic. I am also partial to Michael Gordin's Red Cloud at Dawn which is at times a comparative analysis of the Manhattan Project and the Soviet Atomic Bomb Project, highlighting the similarities and differences, as well as discussing the role of espionage in the Soviet project in a very careful way.