I'm an English speaker of English and some of us have a similar style of speech. Without the American accent obviously. It's generally called RP (Received Pronunciation) or "strangulated vowels". I don't speak it myself but it involves tightening the muscles in the throat during speech. It doesn't come from any part of the country - the entire point is to avoid any regional accent and instead prove one's membership of an elite tribe.

Our royal family evolved a particular style of speech for unknown reasons (many of them were not native English speakers) and people copied them in order to appear socially superior. It's still fashionable today amongst a few social-climbing wannabees but slowly dying out. Up until the 1980s or so the ability to speak that way was absolutely essential to get a job in radio/TV presentation, just like the narrator in the fitness video. Thankfully not so much these days.