I also watched The World at War series. But I have no recollection hearing about the various subterfuges used to fake out the Nazis grouped under Operation Bodyguard, including the network of double agents. I began reading up about it and was fascinated to learn of Joan Pujol Garcia.
Joan Pujol Garcia (Catalan; Spanish: Juan Pujol García), MBE (14 February 1912 – 10 October 1988) deliberately became a double agent during World War II, known by the British codename Garbo and the German codename Arabel. Pujol had the distinction of being one of the few people – if not the only one – to receive decorations from both sides during World War II, gaining both an Iron Cross from the Germans and an MBE from the British.Pujol had a network of spies, all entirely fictional.
The Nazis were completely taken in by Pujol and his network. So much so that:
On occasion, he had to fabricate reasons why his agents had failed to report easily available information that the Germans would eventually know about. For example, he reported that his (fabricated) Liverpool agent had fallen ill just before a major fleet movement from that port on the north-west coast of England. The illness meant that the agent was unable to warn the Germans of the event. To support the illness story, the "agent" eventually "died" and an obituary was placed in the local newspaper as further evidence to convince the Germans, who were also persuaded to pay a pension to the agent's "widow".
Pujol wasn't the only double agent. In fact it appears that the entire German spy network in Britain were double agents.
A decent documentary from the BBC in the topic can be seen on Youtube.