[The Squatter of Amon Rûdh]

This is the sort of problem that Tolkien loved and would have been better able to solve than I. Since he's no longer with us, though, I've done my best with what I have at my disposal. I've found two dates on the internet for the first use of 'twerp' or 'twirp': 1923 and 1874; but most of the sources suggest that it originated in the 1920s. My concise Oxford doesn't even give the correct definition and follows up what it does give with a question mark, but no dictionary I've found gives anything other than 'etymology obscure'. Perhaps the full Oxford would be more help.

Edit: Temporarily I have free access to the on-line OED, which has yielded the following on 'twerp'.

Its first recorded use in a published work is its occurrence in Soldier and Sailor Words by Fraser and Gibbons in 1925. 'Etymology uncertain' is right, though: a quotation from Tolkien's letter appears under the OED entry along with another from R. Campbell in 1957, both of which claim that the term originated in Oxford in reference to T.W. Earp. I would guess that in this case the academics are mistaken, and that this is one of many examples of military slang entering everyday English.