This is a very interesting question you raise here because accurate knowledge of nuclear forces among both sides in the Cold War played a crucial role in the Cold War dynamic.

To answer your question, yes and no. The Soviets stunned the West with their sudden detonation of a nuclear bomb, but it would still take some years for them to build reliable delivery systems, such as ICBM's and Medium range missiles. Throughout the 1950's, the USSR would have to rely largely on its strategic bomber fleet to deliver nuclear strikes, and it was doubtful whether these could reach the US. The USSR developed several hundred medium range missiles aimed at targets in Western Europe to maintain a credible deterrent.

Although these missiles are difficult to hide, for they require launch platforms, it would take some time for the US to develop the intelligence capability to observe and locate such weapons on Soviet territory. In 1956 president Eisenhower authorized the CIA to conduct regular spy flights with the new U-2 plane through Soviet air space to take pictures of military sites on the ground. The Soviets were aware of these flights, but lacked the means to take the U-2 down due to its high altitude. Several attempts were undertaken, and in 1960 they succeeded with pure luck to take one down.

By that time, president Eisenhower had seen enough pictures to conclude the Soviets were not as advanced in missile development as was feared. US politicians, most notably John F. Kennedy, whipped up fears of a missile gap after the USSR beat the US in the space race with sending the Sputnik into space. By that time however, the US had already been massively expanding its own missile production and deployment.

In 1960 the US also launched its first spy satellites, allowing America to regularly monitor Soviet military sites. It is because of these satellites, and information leaked to the CIA by GRU officer Oleg Penkovsky that Khrushchev failed to conceal his attempt to ship nuclear missiles to Cuba. If he had succeeded, America would have been within Soviet nuclear reach for the first time in the Cold War.

In the second half of the 1960's, the US (and also Soviet Union) were so certain of their capabilities to monitor the nuclear arsenals of the other side that this paradoxically generated a sense of mutual confidence. It opened the door towards nuclear arms limitation treaties and both sides knew they could verify whether the other side was keeping to the agreement. The first major agreements on strategic arms limitations were implemented in 1972.

So to conclude, no in the 1950's the West had no idea of Soviet nuclear missiles. Until the 1960s, "UK and US intelligence communities were still uncertain whether the Soviet forces had hundreds of operational ICBM's or none at all", to quote Michael Herman, a scholar and Cold War intelligence veteran from GCHQ. During the 1960's this changed and it opened the door towards nuclear limitation treaties in the 1970's.

Consulted literature:

Christopher Andrew, The Secret World: a History of Intelligence

William Taubman, Khrushchev: the Man and his Era

Michael Herman, "What difference did it make" in Intelligence and National Security, Vol. 26, No. 6, december 2011