I was reading about "grading" and "non-grading" adverbs, came across with the adverb dreadfully. Surprisingly, I found that there are two sources mentioning the adverb as chiefly British , . However, there are other sources that do not mention similar thing. That motivated me to ask about this adverb.
Your second source, the Oxford dictionary (online) says 'chiefly British'. That is its UK version. Its US version does not say 'Chiefly British'. Which version are you goin to believe? By the way, I don't see what your first reference is pointing to, so that it says the word is 'chiefly British'.
Dreadfully can mean terribly, as in I'm terribly/dreadfully tired.. It is also synonymous with awfully and horribly. None of these words would seem overly strange to speakers of American English.
The American Heritage Dictionary lists one definition of dreadfully as
Extremely; very: was dreadfully ill; a dreadfully boring book; dreadfully exciting.