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The informal phrase in two shakes of a lamb’s tail, and its variants, mean: very soon or very quickly.

This phrase alludes to the friskiness of lambs. This is illustrated by the use of the phrase shake of a lamb’s tail to humorously designate a second (i.e., one sixtieth of a minute of time) in the following from the Public Ledger (Philadelphia, Pennsylvania, USA) of Saturday 13^th November 1841 [page 2, column 1]:

Don’t Want Watches.—In the Hoosier State they measure time by the growth of the pumpkin vines, which grow just five feet to the hour, or one inch in every minute.—Exchange paper.
[…] Watching the growth of pumpkin vines to measure the time of day! A watch like that would lose time altogether. Better bestow labor and strength of land upon something more valuable. For instance, grow pork and mark the hours by the joints in a pig’s tail. In New Hampshire they count the seconds by the shakes of a lamb’s tail.

These are, in chronological order, the earliest occurrences of the phrase in two shakes of a lamb’s tail and variants that I have found:
—Note: This phrase seems to have originated in American English. However, quotations 4, 7 & 8, below, indicate that this phrase may have independently developed in Irish English as: in the shaking of a lamb’s tail, also in the wagging of a lamb’s tail:

1-: From the Black Rock Advocate (Black Rock, New York, USA) of Thursday 30^th June 1836 [page 4, column 2]:
—Colonel David ‘Davy’ Crockett (1786-1836) was a U.S. frontiersman and politician who became a legendary figure:

Col. Crockett in a quandary.—Speaking of the great difficulty of always being on the right side, and the danger of non-committal, ‘I never was,’ says the Colonel, ‘in a quandary but once.’
‘During electioneering campaign for Congress, I strolled out in the woods, so much bewildered with politics, that I forgot my rifle. The first thing that took my fancy, was the snarling of young bears which proceeded from a hollow tree, the entrance being more than forty feet from the ground. I mounted the tree, […] my hands slipped and down I went, more than twenty feet, when I landed amongst the family of young bears. I soon found that I might as soon undertake to climb the greased end of a rainbow, as to get back, the tree being too large and smooth. […]
‘[…] I heard a kind of scratching and growling above me, and looking up, I saw the old bear coming stern first upon me. My motto is ‘go ahead.’ As soon as she came within my reach, I seized her tail with my left hand, and with a small pen-knife in the other, I commenced spurring her forward. I’ll be shot, if ever a member of congress raised quicker in the world than I did. She took me out in the shake of a lamb’s tail.’

2-: From the transcript of a speech delivered by one T. Corwin at a meeting held at Carthage, Ohio, on Saturday 13^th July 1844, published in the Carroll Free Press (Carrollton, Ohio, USA) of Friday 16^th August 1844 [page 1, column 4]:

There are salmon in Oregon that jump ten feet out of water, and run up a perpendicular waterfall in the shake of a Lamb’s tail.

3-: From the Tippecanoe Journal (Lafayette, Indiana, USA) of Thursday 31^st July 1845 [page 1, column 7]—reprinted from the Kosciusko Republican:

The town of Monoquet stands on what was once called “Monoquet’s Reservation,” upon the banks of the far-famed Tippecanoe river. Had we the Enchanter’s rod, or the Prophet’s power to call back to life the old war-worn Pottawattamie Chief who sleeps beside yonder stream, he […] might point you to many a spot along these now peaceful shores where were once enacted many a tragic scene of border strife—where the tomahawk and scalping-knife gleamed in the moonlight, and the death-peal of the rifle rung out upon the evening breeze! Poetry in it? He could set your poetic souls on fire; and we—if we had ever courted the sacred muse—could now string out a yard in three shakes of a lamb’s tail.

4-: From The Dublin Evening Herald (Dublin, County Dublin, USA) of Monday 14^th May 1849 [page 3, column 6]:

[In] the Weekly Advertiser of the 5^th inst., […] a provision vendor announces an opening for “a young man, to conduct the butter, bacon, cheese, and ham department, who must have first-rate abilities!”
We can well imagine the chancellor of the exchequer eagerly snapping up the lucky proprietor of such gifts, and popping him into a warm berth in the mint, in the wagging of a lamb’s tail.

5-: From the Springfield Daily Republican (Springfield, Massachusetts, USA) of Friday 29^th August 1851 [page 1, column 1]:

“Talk about your fast horses!” said Jonathan. “Why, I guess you never heerd tell how father’s old blind mare got both her eyes knocked into one, did yer? Wall, you see, we’d turned her out to pastur, one day, in a ten acre lot, where the hornet’s nests was as thick as fifteen gals at a quiltin’ party. And what does the tarnal contrary old critter do but jest go and stick her nigh fore foot right slap down inter one on ’em, which as a nateral consequence fetched the whole etarnal swarm of yaller-legged varmints about her ears, in less’n half a shake of a lamb’s tail.”

6-: From The Emigrant’s Guide to the Western states of America; Or, Backwoods and Prairies (Edinburgh: Oliver & Boyd, [1852]), by John Regan [Chapter 21: Debate on Slavery, page 148]:

“You’re barking up the wrong tree, sartin, if you mean to help yourself out of difficulty by sich talk as that. You’ll get the deeper bemired, and founder right off, if you don’t make off for a neater road nor that ’ere. […] You’ll run against a snag in about three shakes of a lamb’s tail. I can see it.”

7-: From a letter by ‘Terry Driscoll’, dated Stoneybatter, Thursday 9^th September 1852, published in The Clonmel Chronicle, and Southern Counties’ Agricultural and Commercial Gazette (Clonmel, County Tipperary, Ireland) of Wednesday 15^th September 1852 [page 4, column 4]:

There’s no harm in life in residin’ in a counthry where you can speak yer mind on letther-paper, without bein’ warned, in something sthronger than a ‘pig’s whisper,’ that if you do so again, you’ll have to make yourself scarce in the waggin’ of a lamb’s tail!

8-: From a letter by ‘Terry Driscoll’, dated Stoneybatter, Saturday 24^th March 1853, published in The Dublin Evening Herald (Dublin, County Dublin, USA) of Monday 26^th March 1853 [page 4, column 6]:

He invited ’em into a back parlour, and begged they’d wait till he’d send off a bit of a note to a friend that he expected would arrange it for him. The messenger was back in the shakin’ of a lamb’s tail.