The Soviet atomic weapons programme began as early as 1940, after research publications and conversations in the international scientific community alerted them to the probability that the western powers were working on atomic fission and potential weaponisation. From then on, infiltration of the Anglo-American nuclear research programmes was a major strategic priority for both the NKVD (which later became the KGB) and GRU (military intelligence).

In September 1941, John Cairncross — one of the Cambridge Five, a network of NKVD agents working at the highest levels of the British government and intelligence services — alerted his NKVD controller, Anatoly Gorsky, to a meeting of the British government's Scientific Advisory Committee at which the decision was taken to pursue an atomic weapons programme. In June 1942, the Soviets learned of the establishment of the MANHATTAN project. As it became clear that both Britain and America were developing atomic weapons, Soviet intelligence stepped up their efforts (already well underway) to recruit agents inside those scientific establishments.

In late 1941, the GRU recruited Klaus Fuchs, a German communist and physicist working on the British atomic weapons project, codenamed TUBE ALLOYS, at the University of Birmingham. In late 1943, Fuchs was transferred to Los Alamos National Laboratory, the scientific headquarters of the MANHATTAN project, as part of the British delegation.

Fuchs was perhaps the most important of the Soviet atom spies, but he was just one part of a broader network. Other Soviet agents within MANHATTAN included US Army sergeants George Koval and David Greenglass (whose eventual arrest also resulted in the trial and execution of his sister Ethel Rosenberg and her husband Julius) and the American physicist Ted Hall.

Back in Britain in 1937, the NKVD had recruited Melita Norwood, a secretary in the British Non-Ferrous Metals Research Association; in 1941, she was reactivated and responsibility for running her handed over to the GRU; later in the war, her job gave her access to TUBE ALLOYS material which she passed to the NKVD (by then renamed to NKGB.) The GRU also recruited the British physicist Alan Nunn May, a TUBE ALLOYS/MANHATTAN scientist working in Canada.

Most or all of these agents were eventually compromised by defectors or the VENONA decrypts, but not before the damage was done. There are also other agents known to have provided information on MANHATTAN and TUBE ALLOYS, but whose identities remain unclear. The precise impact of the atomic spies on the speed of Soviet atomic weapons development is still debated — but it's something of a moot point. Certainly, information provided by agents within the US and UK alerted Stalin to the danger of a nuclear gap, and encouraged him to make developing the bomb a priority in order to make that strategic imbalance as short-lived as possible.

The hard part about the US making the bomb was not that it was so hard to do. It was that 1. it wasn't clear whether it could even be done, much less which methods would work, and 2. they were trying to make the bomb in a very limited amount of time (around 2.5 years total since they started a true production project). This is why it was a difficult thing to do during World War II, and is still rather amazing: no country other than the US has built their first bomb so fast.

The Soviets had a lot of information on the US project, both in the form of spies and information published after Hiroshima and Nagasaki, and so they knew that it could be done (as was obvious from Hiroshima itself) and had some idea of the different ways that would probably work (they knew which methods the US had used successfully). They didn't use all of that information as efficiently as they could have, because they didn't trust it all (they used it as a "check" against the Soviet scientists' work, because the head of the Soviet project — Beria — didn't trust them either). But it did help guide their research (though it did not stop them from looking into blind alleys, etc.), and give them a perspective on how many resources would be required to be successful.

They threw about the same number of resources at it as the US did — around the same number of workers (500,000-600,000 or so), mostly focused (as with the US project) on the development of fissile material resources.

They took about 4 years to pull it off, starting in earnest soon after Hiroshima. The main constraint on the time was the acquisition of sufficient quantities of uranium ore and the construction necessary to develop the enrichment/production plants. The USSR was much more uranium-poor at the start of their project than the US was at the start of theirs, in part because the US got a boost of ore obtained from the Belgian Congo at the beginning of WWII.

As an aside, Einstein did not work on the Manhattan Project. But plenty of other important scientists did — Fermi, Bethe, Von Neumann, Feynman, among plenty of talented others. The Soviets also had good scientists, though — Kurchatov, Sakharov, Flerov, Khariton, Zel'dovich — these guys were no slackers.