Hm. I wouldn’t have thought this was a UK -> US phrase; I actually thought the opposite. But linguist Geoffrey Nunberg calls it a Britishism, so I guess he must be right. That article was written in 2012, and his annoyance sounds pretty fresh, so has it not been in use for very long there? Or perhaps he always sounds freshly annoyed.
When I looked up the word “will” in the OED, I found their first example of “Will do” dated 1955:
1955 W. Tucker Wild Talent xvi. 217 ‘Paul! Bring my gate pass.’.. ‘Will do.’
But they also link the related military term “wilco,” a portmanteau for “will comply.” There is certainly a military feel about “will do” as well. The earliest citation for wilco is earlier than 1955:
1946 F. Hamann Air Words 56 Willco, will comply; will do.
So I suspect that “will do” sprang from that. But when it entered American usage (your actual question) might be very difficult to determine. It’s a difficult phrase to thrown into an ngram, for example, because it would have been used in many contexts that are different from the one that you are asking about.