First, how do the Soviets get the computer technology to make a Lunar Rover? This was a big reason why the US chose to land on the moon (which was otherwise a pretty random idea). They wanted to pick something that was highly technical in many respects so as to overwhelm Soviet scientific capabilities. So, I'd be REALLY interested in any plausible way the Soviets can actually make a working Lunar Rover.
As for the actual rocket, the Soviets with a WW2 era POD could have got it done. As is, by 1974 The Soviets perfected the Rocket Booster which is now TODAY the premier on Earth. The Energeia (decades later) is still today the best rocket for bringing cargo into space and it has not been flown since 1991. The russian rockets are more efficient and powerful by far. A POD which significantly saves Soviet GDP and perhaps gets Von Braun captured=the capaibility to go to the moon no problem. The real issue is landing on it.
But presuming the PODs work where Russian lands one month later, the result is the Soviet Union gets to Mars first, without a doubt, by the mid 1980s. I do not doubt this. The Soviets had superior space stations and rockets, they'd be able to get far more cargo to the moon faster and into orbit. The American approach to rockets was fundamentally misguided and they would not be able to catch up to the Russians without substantial espionage. Ironically, the Soviet system worked for the same reason the Sputnik V is probably the best Coronavirus vaccine today. Russian science is ruthless. It operates based upon catastrophic failure. Watching something blow up and fail is more efficient, and dangerous, than using a dyno. The Soviets did not have rocket booster dynos. They had to watch their crap blow up. They also had to build their rocket boosters to fit on a train on the transiberian railroad. The US used dynos and used boats to transport components. Long story short, necessity is the mother of invention and Soviet rocketry punched way above its weight and it was the only thing the Soviets developed that was all around superior to the United States during the Cold War.