Living in the US, I only ever remember hearing the word “cheers” in the specific context of a champagne toast. Living in the UK, I believe there is no word that I hear uttered more times each day than “cheers.” A quick search on Google would indicate that I am not alone in noticing this. I would understand the phenomenon better if the word was used to replace some other common words, such as “thanks” or “goodbye” but instead it appears to me that it is added to get constructions such as, “thanks, cheers” and “cheers, goodbye” which I find slightly mystifying. Even more unusual is the repetition of the word in a single closing utterance, along the lines of “cheers, goodbye, cheers.”
Regardless of my feelings about the numbing repetition of this word in casual conversation, as the “American’s Guide to Speaking British” points out, it’s impossible to pull this one off in an American accent. We’ll have to stick with the classic “toast” meaning and find another way to exit all conversations.