This doesn't happen. One's native accent is determined either by conscious adoption of certain features or by subconscious imitation of the social group that one is most associated with. This is why you will find that children talk like their parents until school and as teenagers sound more like each other than they do like their parents. It's why half the military sound they're like from North Carolina (the other half being from South Carolina).
His tick is a faux Brahmin accent ("rawther" for "rather," "nawt" for "not," and especially "praps" for "perhaps"), which used to be a dialect that I refer to as American Common Academic. He is not the only BYU professor who spoke that way (I can think of one another, in Anthropology, though he had no connection to Mormon apologetics other than contempt for it, so I hesitate to drag his name in here), but it certainly was the kind of accent many people adopted, consciously or not, in spending most of their time around other academics. This is not at all an unusual phenomenon. His academic accent seems strange because he speaks an archaic variety of this dialect, while contemporary ACA is more or less colored by features traditionally thought of as "feminine" and associated with the "Valley Girl" accent that arose the in the late 1970s; it eschews features associated with the Dead White Male Brahmins. He sounds like the careerist academic from the late 1970s that he is (see also, Frasier Crane).
What is hilarious is that he claims to have lost his "R" due to his perfect Standard German. Standard German that you learn in schools or colleges in the US and Britain does drop its uvular "R" in certain positions (mostly at the codas of words and syllables), which British English does in its south eastern quadrant and certain dialects of American English do in the northeast and (formerly but no more) in the Deep South.
How he picked that up from Swiss German is a mystery, though. The "R" of Swiss German which he would have been immersed in for two years uses a trilled "R", not the uvular "R," and it doesn't have the R-drop feature (unlikely in the 1970s anyway). Even the "Standard German" in the media and educational settings there use a trilled "R." The uvular "R" of Standard German has been spreading in Switzerland, but only really since the late 1980s, which would have been after his time there. In short, he wasn't in the right German speaking country at the right time to pick up that still rather restricted feature. He should be going around trilling all of his "Relief Society" or dropping them at syllabic codas in fast speech.
Funniest of all to me: perhaps it was an effort to humor his host when he accepted that his accent has some vague "East Coast" quality to it, but he doesn't even drop his "R" in his speech!
One wonders why, with a Standard German so perfect that he would have been mistaken for an actual German, he says that Swiss people in the 1970s thought this young man to have been a former Nazi, presumably born in the 1920s or 1930s, rather than just an ordinary young German (see 9:30). I am tempted to suspect that his off-the-cuff attempt to impress the host is being influenced by the apocryphal story about Hugh Nibley, mentioned by Gadianton.
That story itself is utter BS, of course. Hugh Nibley dropped plenty of German in his lectures, always spoken with an accent as atrocious as any tourist from Utah trying to buy "Fawrkarden" at the "Bawnhawf." His Arabic accent was also terrible, and on an old message board his son-in-law Boyd, who has a graduate degree in French, wrote that, while Nibley's understanding of French was probably better than his own, his pronunciation of French was abominably bad. He used to read scholarly articles, so why would it be otherwise? It therefore seems very unlikely to me that Hugh Nibley ever impersonated a German-speaker during the Second World War.