The term "Cold War" was used to describe the emnity and tension between the two competing systems of communism and capitalism after the Second World War that never actually resulted in direct warfare - although several "hot wars" were fought by proxy forces armed by both sides - Vietnam, Angola, Afghanistan, Nicaragua to name several.
The UK, or to be exact the British Empire, was intially seen by the Soviet Union as their number one threat after the revolution of 1917, as the 20th Century progressed the USA was the Soviet's main concern. The UK initially saw itself as a third way between the USA and Soviet Russia - Britain in 1945 had a socialist Labour government - but following events such as the Soviet Berlin blockade, tied in their lot with the USA. Post Suez, Britain's foreign policy was based on their "Special Relationship" with the US.
Britain backed the USA in the UN Security council, send troops to Korea (but not Vietnam) and had a cooperative relationship with US intelligence agencies (in John LeCarre's novels "the Cousins") sharing intelligence on Soviet Union and its allies.
Britain's intelligence services were focused on the Soviet threat during the Cold War, but life in Britain for the general public was not affected by the Cold War anymore than in the USA, apart from the omnipresent threat of nuclear annihilation.
The Arms Race refferred to the race between Russia and America to arm themselves with ballistic nuclear weapons, the Cold War refers to the period after the Second World War to the fall of Communism - although this term simplifies 60 years of international relations.