In American English, cockerel applies to the young male (Webster's Collegiate). The OED also applies it to the young. Thus it is not strictly equal to rooster, an adult male.

Forgotten is H. L. Mencken's "The American Language, An Inquiry into the Development of English in the United States." First published in 1919, I have the 4th ed., 1937; haven't checked to see if there were later editions. At the time it was fairly comprehensive.

Of course the two languages developed many more words, mostly slang, as time went on, but now, with TV and ipod and all that, newer words are picked up in UK, Canada and U. S., etc., pretty much at the same time.
I doubt that any scholars now seriously speak of two languages.

Many Americans and Canadians listen to the BBC news on cable television. One distinction is the treatment of certain collective nouns:
English- Army are, American- Army is. Etc.