I notice that the use of the word "sure" to mean "yes" seems to be much more common in American English than in other dialects. Can anyone point to any evidence as to the origins of this divergence?
The AHD says its usage dates back to the mid 19th century. The expression is probably the short of sure thing, a common Americanism, as from the sources shown below:
Yes; certainly : Sure, I'll support you (1842+)
From “Eight ways to say yes” Sure:
Sure is related to the Old French word of the same spelling which meant "safe" or "secure." The word was used to mean "safe" until the early 1500s. In the mid-1500s, sure started to take on senses of "certainly" and "indubitably," which helped introduce popular phrases such as "to be sure" and "for sure" to English speakers. The common Americanism "sure thing" arose in the 1800s.
This seems to be a perfect situation for using Google nGrams.
First, etymologically, sure as yes could have come from many sources: 'surely', 'sure thing', 'sure enough', 'to be sure', 'for sure'. The use of a word like 'sure' for an enthusiatic affirmative or 'certainly' is common in many languages: GE 'sicher', FR 'bien sûr', IT 'certo', RU 'наверняка́', so it can't be considered an English, much less an American English, anomaly.
Etymonline states that "The use as an adverb meaning "assuredly" goes back to early 14c." well before the American colonies were a twinkle in the eye of the English colonizers.
But now to nGrams: just looking for the simple string 'sure', it does seem there is in written language a noticeable change in its general frequency of use starting in the 1970's:
But ngrams is limited in that it only searches for instances of the string and doesn't take into account context. There's 'sure fire', 'make sure', 'as sure as', 'be sure', etc all sorts of collocations where it is not just 'sure' as a single word affirmative. Looking at the results of nGrams is a good way to find these contexts.
The likely context that you care about is 'sure' at the beginning of a sentence or even stand alone.
Google nGrams is case sensitive and has a marker for sentence beginnings, so 'Sure', for the beginning of a sentence, shows:
where there is this similar rise after the 70's in US English (but flat in the UK). But there's also the anomaly of a big bump in the US in the 30's and 40's.
A common if colloquial and regional variation is 'Sure enough':
which has similar shape to 'Sure'
After all this, searching through the hits that google returns, there is no apparent incident that is obvious that accounts for the rise in the use of 'Sure' as affirmative in the US after the 70's.
But your intuition is confirmed that there is a difference between the US and UK for 'sure' as an affirmative.