So I've just watched the programme. Unfortunately there's nothing in english or with subtitles, but for german speakers, here's the link:
Hitlers Geheimwaffenchef - History - ZDFmediathek - ZDF Mediathek
It seems that the theory that Kammler's survival is more than just another conspirational rumor - which a programme of the rather conservative (not privately owned) TV channel would avoid like the devil or, if at all, make fun of - seems to be based mainly on the statement of "whistleblower" John Richardson, who says that his father, Daniel W. Richardson was responsible for snatching Kammler away from the Soviets and getting him overseas. Richardson (father) worked for the CIC and seems to have been a man for the "top secret" missions who would take orders from Eisenhower and Roosevelt directly. According to his son he said in private conversations that he had interviewed Kammler after the date of his supposed sucide (May 9th 1945) and seen him off to the US.
John Richardson apparently made this statement in an Austrian documentary called "The Devil's Treasure". I can't find anything online, so I guess it's possible that this hasn't been released yet. Both the documentary film maker and John Richardson seem to be legit and not of the right wing / conspiracy nut persuasion, but of course, it's hard to tell just from the look of them. It's obvious, though, that Daniel Richardson really was a high-ranking secret service man.
If anyone has information about the Richardsons, please share.
There are a few other sources, though, that have "recently been uncovered", as the history programme says.
In 1949, the American government of Hessia (Germany) ordered a special investigator to find out what happened to Kammler. His report states that the CIC quite freely admitted to having arrested and interrogated him but that he later managed to escape, just to be subsequently caught by the Russians. A historian in the TV programme thinks that the latter part was probably a cover story by the CIC. The special investigator's report concludes that Kammler was probably still alive and that his suicide had been fabricated. There were at least six different versions of his suicide, but his body was never found.
Another source from Austria is mentioned, but with no more information than that "it has to do with banking".
Nothing is said about where Kammler is supposed to have lived in the years after the war, which projects he was involved with or what knowledge he imparted. The programme says that his main "working areas" at the end of World War II had been rockets and jetplanes. A third area is mentioned but the documentary kind of dances around the subject, indicating that it probably was nuclear bombs.
It seems that in the interview for the Austrian documentary, John Richardson says that his father told him Kammler did commit suicide in the end. After only a few years in the "isolation" of his new life, he supposedly hanged himself. Again, no further details are given. I'll probably have to wait until I get to watch this "Devil's Treasure" film and listen to the interview with John Richardson to learn more.