Dictionary coverage of 'get cracking'
J.E. Lighter, Random House Historical Dictionary of American Slang (1994) reports that "get cracking" came into U.S. English from the UK during the 1940s:
get cracking to get busy; get going. {This phr[ase] came into U.S. speech through contact with British armed forces during WWII.}
Lighter's first citation for the phrase is from Eric Partridge, Dictionary of Slang and Unconventional English (1936), as "Get cracking, begin work."
It appears to have caught on as a naturalized phrase fairly quickly, however, since it appears in Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960):
cracking, get Start; start moving; begin working; begin to exert oneself. Often in the phrase let's get cracking.
The entry for "get cracking" in the original edition of Eric Partridge, Dictionary of Slang and Unconventional English (1936) is actually a bit more detailed than Lighter's citation suggests:
get cracking. To begin work : Royal Air Force : from ca. 1925. I.e. cracking on speed.
Earlier in the same dictionary, Partridge has this entry for cracking as an adjective:
cracking, adj. Very fast ; exceedingly vigorous (-- 1880) : slightly ob[solete, in 1936]
By the fifth edition (1961) of the dictionary, however, Partridge had evidently rethought the place of "get cracking" in the larger scheme of similar phrases and had adopted the entry that ghoppe reports in a different answer here, with the suggestion that it might have its origin in "whip-cracking at the mustering of cattle."
Early Google Books matches
Google Search results generally support Partridge's1936 reading of the phrase. The earliest relevant matches are from 1938, and one of them is RAF-specific. From Popular Flying, volume 6, issue 9 (1938) [combined snippets]:
We've got a new vocabulary in Black Bourton, the vocabulary of the R.A.F. Nice things are "wizard"; nasty people are "ropey types," and a reckless man is "split." One is "browned off" when one is rather depressed; the ground is the "deck" and the sky is the "ceiling"; dull-witted people are "drips," and when in a hurry we ask each other to get "cracking." This strange lingo is, of course, subject to changes and ex-R.A.F. men of several years ago would find themselves hopelessly dated by the expressions they use.
There is, however, one outlying instance of "got cracking"—or more precisely "got cracking away"—that might be significant as well. From Bertram Milford, In the Whirl of the Rising (1904), a novel seemingly set in Rhodesia during the Second Matabele War, of 1896–1897:
"You must have had the very devil of a scrap, Peters." one of these [horsemen from Green's Scouts, a relieving force] was saying. "We could hear you banging away from the time you began, and pushed our gees for all they'd carry; for we reckoned all that shooting meant a big thing and no bally skirmish. ... They [the Matabele forces] didn't know we were there till we got cracking away right in their faces, or mostly backs. Magtig! Didn't they skip. ..."
Here it seems pretty clear that "got cracking away" means "started firing our guns." Is this usage connected to the later World War II use of "get cracking"? It's difficult to say. If there is a connection, the literal slang term for gunfire would have to have transmuted into a more figurative sense of getting started. Still, both usages have a strong tie to military usage, and a connection is not impossible.
Conclusion
No reference work has convincingly tracked "get cracking" to its original lair. The term does seem to be of UK English origin, but whether it originated in the RAF in 1925, or in Rhodesia in the late 1890s, or in some bucolic setting to the sound of bullwhips bring cattle into order for transportation or migration remains unclear.
Best Answer
New to me, but the OED gives it as US slang and from the verb ‘lam’, meaning ‘to run off, to escape’, which, again, is US slang. The origin sems to be in an Old Norse word which is cognate with ‘lame’.