Thank you very much, indeed (TYVMI) has long been a go-to phrase for British interviewers and interviewees. How long? Well, it has been afoot at least since 1973, when Anthony Burgess made this amusing observation in the New York Times:
British gabbiness is also to be associated with a kind of obliquity or indirectness, which is meant to be polite, though sometimes it can be as cold as silence. Thus, an American says, “Have you change for a ten?” but an Englishman will say, “I’m really most terribly sorry to bother you, but I don’t suppose by any chance you might have such a thing as change for a pound, would you—the old quid, you know? Oh, you would? I’m really most terribly grateful. Thank you, you’re an angel. Thank you very much indeed.”
Fifteen years later, Times TV critic John O’Connor wrote this about David Frost’s substitute hosting duties on the “Today” show:
Mr. Frost is more of a bon vivant (than Jane Pauley), never at a loss for an amusing anecdote and, even through a certain early-morning bleariness, always maintaining a remarkable enthusiasm. ”Wonderful stuff!” or ”Thank you very much, indeed,” says Mr. Frost at regular intervals.
For some time, I have been waiting for TYVMI to emerge from a pair of American lips. I believe I have heard a couple of NPR hosts say it, but I didn’t take notes so can’t be sure. Christine Amanour of CNN and ABC and Stuart Varney of Fox say it all the time, but they are Brits. I will stay on the lookout, but for now have to content myself with one American sighting. It was uttered in May of this year by Paul Schott Stevens, a native of New Orleans and president Investment Company Institute, at the close of a conversation with Alan Simpson and Erskine Bowles. If by any chance you want to hear it for yourself, be my guest.
notoneoffbritishisms.com 
I really don’t think “Thank you very much, indeed” is going to jump the pond. If you were to do a search of the corpora you might find a few examples, but it is so idiosyncratically British that it might as well be French or German. It fits exactly into the the English manner of indirection, exemplifying a national linguistic characteristic, reminiscent of pottering about, dithering and fussing. It’s an endearingf turn of phrase, but distinctly un-american.
Very nice, except: how many Americans would utter “have you any change”? Maybe the same ones who follow thank you very much with “indeed”!
Absolutely right. AE=”You got change for a ten?”
“Thanks muchly” is an expression that has gotten me rolled eyes at home in the US, and mild surprise in Yorkshire, where they seem to think it’s specific to Yorkies.
Or alternate AE: can you break a ten?
I’m from Dorset, and I picked up ‘ta muchly’ from somewhere, but I don’t think it’s all that common. I live in London now, and I never hear it.
I have a feeling that it’s a Brummie expression. Beryl Reid used it as her ‘Marlene, the girl from the Midlands’ character back in the 1950s/60s. The similar ‘Ta everso’ sounds Scouse to me.
Just discovered this blog – excellent! Being a native south-east Englander, another phrase I find myself using a lot is “thanks ever so much”, which looks quite pretentious written down, but sounds quite normal when spoken. It is quite formal, I use it to round off paying the bill in a restaurant for example, or on the phone to the bank. It’s always struck me as very British
I would venture to say that no American has ever said, “Thanks ever so much.”
On BBC radio 4 effete southern weather presenters in particular say “TYVMI” when they are simply handed their cue to give the forecast. A gross over-reaction. If there were more Yorkshire folk presenting the forecast you’d be lucky to get a “Ta”!