Extraordinarily well, at least by comparison with German espionage efforts in the same period. By 1941, the German intelligence network within the UK was comprehensively compromised by MI5 — the Double-Cross System is still regarded as one of the most strikingly successful pieces of counterintelligence work ever.
By comparison, the Soviet intelligence services in the 1930s and 1940s succeeded in penetrating the British military, intelligence and scientific establishments to a genuinely remarkable degree. Most notable are the Cambridge Five, a network of Soviet penetration agents recruited at Cambridge University in the early 1930s who went on to successful careers in British government service:
- Kim Philby: served in the Secret Intelligence Service (MI6) and Special Operations Executive during the war. He did stints as an SOE trainer, as an MI6 liaison officer at Bletchley Park, and as a senior counterintelligence officer. After the war, he worked as MI6 station chief in Istanbul and intelligence liaison in Washington DC. Despite earlier suspicions and investigations, he wasn't conclusively identified as an NKVD/KGB agent until 1962; shortly thereafter, he defected to the USSR.
- Anthony Blunt: served in MI5 during the war, in a position which gave him access to the Ultra intelligence produced at Bletchley Park. After the war, he worked in the Royal Household, including as Surveyor of the King's Pictures (he was an art historian by training.) There were suspicions that he was a Soviet agent as early as the mid-1940s, but he wasn't finally unmasked until 1963 — and in exchange for his confession and cooperation with MI5, wasn't publicly identified until 1979.
- Donald Maclean: a Foreign Office diplomat, working in a senior post in London during the war and as a senior embassy official in Washington from 1944-48; later, he was posted to Cairo and back in London before eventually being uncovered as a spy by the Venona project; in 1951, having been alerted by Philby (who saw the Venona decrypts thanks to his position in DC), he defected to the USSR.
- Guy Burgess: served in MI6 during the war, and later became a Foreign Office official, serving in a variety of roles — including as a ministerial aide. He wasn't revealed to be a spy until 1951 when, while working in the British embassy in Washington (and living with Philby), he accompanied Maclean in defecting to Moscow.
- John Cairncross: employed at the Government Code & Cypher School at Bletchley Park during the war, working on the Ultra project. Later, he joined MI6, and after the war joined the Treasury; he confessed to espionage in 1951, but wasn't prosecuted. His identity as the 'fifth man' in the Cambridge ring wasn't conclusively established until 1990, when KGB defector Oleg Gordievsky confirmed it.
The Cambridge Five are only the most memorable Soviet spies, however — mostly by virtue of their seniority in government service. It's commonly speculated, for example, that Philby, a rising star in the intelligence services, could one day have become 'C', the head of MI6.
But there are others: for example, the Soviets succeeded in penetrating the wartime Anglo-American atomic weapons programmes to a remarkable degree. In Britain, Tube Alloys (the codename for the early British nuclear programme, later merged into the Manhattan Project), included at least three Soviet agents:
- Klaus Fuchs: a physicist originally employed on Tube Alloys work at the University of Birmingham, he later transferred to the Manhattan Project at Los Alamos, and was one of the Soviets' most valuable scientific intelligence assets until his arrest in 1950. Like Maclean, he was identified by the Venona decrypts.
- Alan Nunn May: also a physicist, working on Tube Alloys at the University of Cambridge; he too later transferred to the US to work on the Manhattan Project. His spying was discovered and he was arrested after a Soviet intelligence officer, Igor Gouzenko, defected to Canada in 1945.
- Melita Norwood: perhaps the most striking story, Norwood was a secretary at the British Non-Ferrous Metals Research Association, which was involved in Tube Alloys research during the war. She continued providing intelligence to the KGB until her retirement in the 1970s, and despite falling under suspicion at various points was never prosecuted. In 1999, KGB archivist Vasili Mitrokhin defected to the UK; her identity as a source was revealed in the enormous cache of documents he brought with him. She died in 2005.
All of the agents I've listed are only the most notable and/or interesting. There were many others. The Soviets' success in infiltrating agents in the UK ebbed and flowed over time; they hit a lull after Stalin's purges in the 1930s, for example, and again after the winding-up of the Cambridge ring and the success of the Venona project in the late 1940s and early 1950s. But during the war, Britain was a major — arguably the primary — target for the Soviet intelligence services, and their success in penetrating the British establishment marked a high-point in their history.
For further reading, I recommend:
It's less academic, but Kim Philby's memoirs also make for interesting reading.