Yes and no. It seems that the British did send spies and agents to the United States but it doesn't seem that they were there to spy on the US per se. Rather the British Security Coordination were tasked with bringing the US into the war. Generally speaking the BSCs efforts attempted to 'drag the United States into [the war]' (Winston Churchill to his son) by spreading propaganda explictly aimed at making the US populace want to join a war on the side of Britain. One such example of this is the book 'Sequel to the Apocalypse;: The uncensored story: how your dimes and quarters helped pay for Hitler's war'. The book was written by Rex Stout, George Merton and Sylvia Porter of the New York Post after a BSC agent, Donald MacLaren employed them to do so. It's worth noting here that George Merton himself was a BSC agent.

The BSC contained such members as Roald Dahl and Ian Fleming who - among with the other agents, were tasked with "clandestinely promoting British interests in the U.S. while also working to undermine remaining isolationist attitudes in American politics and society" The BSC was founded in 1940 when Winston Churchill became prime minister and at the time neither Russia nor America were in the war against the Nazis. It therefore makes sense that the Brits would employ agents in an attempt to coax the Americans into joining them in their war against the Nazis.

Sources: On the head of the BSC, On their founding and operations, [Number 4 shows Roald Dahl and Ian Fleming's involvement in the BSC] (http://www.history.com/news/history-lists/6-people-you-didnt-know-were-wwii-spies), The aforementioned propaganda book written by the New York Press on behalf of the BSC.

I wasn't able to find anything on the US 'spying' on the Brits or conducting operations on British soil, but according to the CIA's own site the Office of Strategic Services wasn't founded until June, 1942 at which point the British and Americans were firmly fighting on the same side.