The original form for at least some of those words is to start with "hw". For example, the first line of the Anglo-Saxon poem Beowulf is "Hwaet! We Gardena in geardagum". "Hwaet" is the word that became "what" in modern English, and was (given that when the poem was written people tended to spell relatively phonetically) presumably pronounced. According to http://www.etymonline.com/index.php?allowed_in_frame=0&searc... it's from a Proto-Germanic form *khwat.
I don't know when or why the "h" and the "w" got swapped around, though. Certainly, when people pronounce the "h" it it's more like "h-wot" than "w-hot". Actually, perhaps even more like "f-wot".
I guess the question is still open as to whether the people who pronounce the "h" today are continuing an unbroken tradition that goes back to the Anglo-Saxons (and the majority who don't are using a simplified form), or whether the "h" dropped out of use sometime after the spelling was fixed, and modern speakers are re-introducing it.
I can say, though, that it's generally regarded as "better" to pronounce the "h" by a lot of English people, and it's also strongly pronounced in certain Scottish accents. (Some friends from "posh" parts of Edinburgh definitely pronounce it, and also say something like "Wed'n'sday" pronouncing the "d".)
(Aside: I've read that the "what!" at the start of the poem was a use of the word as something similar to "ah!" or "lo!" or some other relatively meaningless poetical interjection. This gives a wonderful historical context to P.G. Wodehouse's characters greeting each other with a "what, what, what, old thing!". Bertie Wooster was apparently using a construction that dated back a thousand years...)
Swedish has dropped the H (along with other differences) giving vad, vem, vilken, var, vita for what, who, which, where, white.
In the golden radio and early tv days, broadcasters and actors learned to speak with a "mid-atlantic" or transatlantic accent that was constructed to sound like something halfway between American and British Englishes. For a while, it was even taught in upper-class private schools in the U.S. North East.
edit: in fact, the whine-wine merger's own page indicates that the only place they're not merged is a strip of the american South.
(Loughborough)