To drop the final shoe from our final discussion, I couldn’t resist turning to the OED to see what it said about “getting the mitten”. Here are the most relevant definitions (click on the picture below to enlarge the text).
So Wodehouse’s usage first appears in 1838; and note how quickly the illustrative quotations start coming from American sources. I also like the earlier (16th-18th century) entries that lay the figurative groundwork for the later colloquialism.
And for those who enjoy browsing the OED, here’s a link to a slightly larger excerpt about “mitten.”