While Britain definitely took the backseat in the Cold War, which was largely fought between the US and the Soviet Union, it still played a pretty significant role in the way things went down. This is hardly surprising, seeing as the UK and Russia have a long (read: looonnnggg) history of rivalry based on ideological differences.
Although UK Prime Minister Winston Churchill had initially worked with Soviet Union leader Joseph Stalin towards the end of WWII to eradicate the Nazis and rebuild Europe, this relationship quickly crumbled. In the aftermath of the war, Churchill adopted a vehemently anti-Communist stance that saw him cooperate with US President Franklin D. Roosevelt and his successor, Harry Truman. By 1946, the pretense of having any civilised relationship with the Soviets was up; Churchill’s “Iron Curtain” speech condemned the Soviet Union’s policies in Europe on 5 March. The speech is largely seen as the most famous oration of the Cold War period, even though Churchill had been voted out as PM by this point.
Churchill’s speech is particularly significant as it had a massive influence on the Truman Doctrine, and later the Marshall Plan, which served to further solidify the UK’s relationship with the US, thus creating a united front against the Soviet Union and the Communism it championed. This is why there is a strong argument for the notion that Britain played a key role in developing ideas that led to key events in the Cold War.
That’s all you need to know in terms of the UK’s indirect influence on the Cold War for now. Directly speaking, however, Britain made a few significant attempts to prevent the spread of Soviet power. Should you want to know, British troops were sent to Greece in 1946 to prevent a Communist seizure of power immediately after WWII. This happened because many of the resistance movements in Nazi-occupied countries had been Communist and they represented a political force in the immediate post-war world. Unfortunately, Britain could not afford to support the fighting against Communists in Greece, and the task was taken over by the United States. This in itself is a comment on the changing roles of the US and the UK as world peacekeepers — but goes to show that the UK was active in the Cold War.