The Oxford English Dictionary’s (OED) definition 8b for the word cheer is: pl. A friendly exclamation or exhortation to be cheerful; esp. a salutation before drinking. The first citation comes from the copy in an advert (sorry) in a 1919 issue of the magazine Sphere: “Cheers—I’m longing to see you and a Kenilworth together—the two nicest things on earth.” The first drinking example given isn’t until 1946–surprisingly late, it seems to me. Another meaning had cropped up by 1976, the date of a citation from The Times: “By a remarkable transition from the pub to the sober world at large outside cheers has become the colloquial synonym in British English for ‘thanks’.”
With all these meanings, plus the plural of cheer, as in fans’ literal or figurative shouts of encouragement, it is impossible to objectively chart the American use of the word, at least with my limited resources and skill. My impressions are: the drinking salutation took off here right away, and persists; the “thank you” meaning (which I remember being startled by in London circa 1996 when a newsagent [ NOOB alert!] took my coin and handed me a Guardian) has not yet arrived; and the “friendly exclamation or exhortation to be cheerful” has been a resounding success in one and only one realm: e-mail signoffs. This is widely practiced and seemingly inoffensive, yet it raises the hackles of some. E.g., cheers bothered a respondent to a Lifehacker survey on annoying signoffs because “the sender is almost never (a) British or (b) sharing a drink with me.”
I will not get into cheerio, except to note that it has not penetrated and surely never will penetrate the U.S., except as something the stage Englishman says, and to guess that at this point, it’s used almost solely ironically in the U.K. That is suggested by the final citation in the OED, from P.G. Wodehouse, in which the word has shifted parts of speech, often a sign of its decadence: “You could not have found a more cheerio butler.”
notoneoffbritishisms.com 
Cheers has long been the most common bar salutation in the U.S. There was even a sitcom named after it, which you may remember.
And “cheers,” meaning “thanks,” goes back many years in the UK.
I don’t understand why your site features such ignorant nonsense. Do you think
extensive quotes from the OED will disguise the fact that you don’t have a clue?
@James Green
Did you even read the post? Do you understand the point of the website? Not content with being needlessly rude, you were needlessly rude from a position of total ignorance.
“Cheers” has also become an email valediction synonymous with “Yours” and “Regards.”
Oops, you mentioned that in your post!
‘Cheerio’ is still commonly used throughout Britain without attempt at irony.
Cheers and cheerio
Ben
I have been hearing “cheers” in America as “thanks” over the last two years or so. Heard it at a bar this past weekend after I paid for my beer. Mind, it was during a morning watching English “soccer”, but the bartender and nearly all the patrons were American.
I began a multi-year interlude in the UK in the mid-80s, when I was in my early 20s. I grew up and was initially educated in the northeastern US, and prior to going to the UK, I had never heard “cheers” in any context other than toasting. I then became somewhat accustomed in the UK to hearing people say “cheers,” meaning either “thanks” or “bye,” but I never said or wrote it in my time there. Around 1994, having been back in the US for about 5-6 years, I began working at a major eastern university, and I was stunned when some co-workers, including some who were then in their late 40s, ended emails by saying “cheers.” Since then, especially in the last 5 years, I have witnessed a slow but steady uptick in the number of colleagues and students who use that closing and other Britishisms (“no worries,” an expression I don’t ever recall hearing in the UK in the second half of the 80s, is perhaps the expression I most commonly hear these days).
‘Cheerio’ used to be used as a pre-drinking salutation but has been supplanted in that role by ‘cheers’. I suggest that the same thing has happened with the ‘goodbye’ meaning. But ‘cheerio’ has never been used to mean ‘thank you’.
Cato, ‘No worries’ is an Australianism gaining ground in the UK, where ‘not to worry’, ‘no problem’ or ‘not a problem’ would be the native equivalents.
Also, there is the greeting “What cheer?”, usually rendered down to “Wotcher?” in fiction, and denoting working-class/cockney characters.